· 82 JANOS KOLL AR [Greenberg-Harper81] M. Greenberg and J. Harper, Algebraic topology,...

JOURNAL OF THEAMERICAN MATHEMATICAL SOCIETYVolume 12, Number 1, January 1999, Pages 33–83S 0894-0347(99)00286-6

REAL ALGEBRAIC THREEFOLDS II.MINIMAL MODEL PROGRAM

JANOS KOLLAR

Contents

1. Introduction 332. Applications and speculations 393. The minimal model program over R 444. The topology of real points and the MMP 485. The topology of divisorial contractions 506. The gateway method 547. Small and divisor–to–curve contractions 588. Proof of the main theorems 629. cAx and cD-type points 6610. cA-type points 6911. cE-type points 7412. Hyperbolic 3–manifolds 78Acknowledgments 81References 81

1. Introduction

In real algebraic geometry, one of the main directions of investigation is thetopological study of the set of real solutions of algebraic equations. The firstgeneral result was proved in [Nash52], and later developed by many others (see[Akbulut-King92] for some recent directions). One of these theorems says thatevery compact differentiable manifold can be realized as the set of real points ofan algebraic variety. [Nash52] posed the problem of obtaining similar results us-ing a restricted class of varieties, for instance rational varieties. For real algebraicsurfaces this question was settled in [Comessatti14].

The aim of this series of papers is to utilize the theory of minimal models toinvestigate this question for real algebraic threefolds. This approach is very similarin spirit to the one employed by [Comessatti14]. (See [Silhol89, Kollar97a] forintroductions to real algebraic surfaces from the point of view of the minimal modelprogram.)

Received by the editors January 26, 1998.1991 Mathematics Subject Classification. Primary 14E30, 14P25, 14E05; Secondary 14M20,

57N10.

c©1999 American Mathematical Society

33

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34 JANOS KOLLAR

For algebraic threefolds over C, the minimal model program (MMP for short)provides a very powerful tool. The method of the program is the following. (See[Kollar87, CKM88] or [Kollar-Mori98] for introductions.)

Starting with a smooth projective 3-fold X , we perform a series of “elementary”birational transformations

X = X0 99K X1 99K · · · 99K Xn =: X∗

until we reach a variety X∗ whose global structure is “simple”. (As usual, a bro-ken arrow 99K denotes a map that need not be everywhere defined.) Neither theintermediate steps Xi nor the final X∗ are uniquely determined by X . In essencethe minimal model program allows us to investigate many questions in two steps:first study the effect of the “elementary” transformations and then consider the“simple” global situation.

In practice both of these steps are frequently rather difficult. For instance, westill do not have a complete list of all possible “elementary” steps, despite repeatedattempts to obtain it.

A somewhat unpleasant feature of the theory is that the varieties Xi are notsmooth, but have so-called terminal singularities.

The aim of this paper is to study the minimal model program for real algebraicvarieties.

Definition 1.1. By a real algebraic variety we mean a variety given by real equa-tions, as defined in most algebraic geometry books (see, for instance, [Shafarevich72,Hartshorne77]). This is different from the definition frequently used in real alge-braic geometry which essentially considers only the germ of X along its real points(cf. [BCR87]). In many cases the two variants can be used interchangeably, but inthis paper it is crucial to use the first one.

If X is a real algebraic variety, then X(R) denotes the set of real points of Xas a topological space and X(C) denotes the set of complex points as a complexspace. XC denotes the corresponding complex variety (same equations as for X butwe pretend to be over C).

For all practical purposes we can identify X with the pair (X(C), complex con-jugation) (cf. [Silhol89, Sec.I.1]).

A property of X always refers to the variety X . Thus, for instance, X is smoothiff it is smooth at all complex points, not just at its real points. We use the adjective“geometrically” to denote properties of the complex variety XC. (For instance,(x2 + y2 = 0) is an irreducible plane curve over R which is not geometricallyirreducible.)

If X is a smooth projective real algebraic variety, then there is a variant of theMMP where the intermediate varieties Xi are also defined over R. We refer to thisas the MMP over R. In developing the theory of minimal models for real algebraicthreefolds, we again have to understand the occurring terminal singularities. Thiswas done in the first paper of this series [Kollar97b].

The MMP suggests the following two-step approach to understand the topologyof X(R):

(1) Study the topological effect of the “elementary” transformations.(2) Investigate the topology of X∗(R).

The aim of this paper is to complete the first of these two steps.


REAL ALGEBRAIC THREEFOLDS II. MINIMAL MODEL PROGRAM 35

It should be emphasized again that although at the end we care only about thereal points X(R), for the proof it is essential to keep track of all the complex pointsas well.

We are unable to say much about this question in general. There are serious prob-lems coming from algebraic geometry and also from 3-manifold topology. Some ofthese are discussed in section 4. Our aim is therefore more limited: find reasonableconditions which ensure that the steps of the MMP can be described topologically.

The simplest case to study is contractions f : X → Y where X is smooth. Over Cthe complete list of such contractions is known [Mori82], and it is not hard to obtaina complete list over R. From this list one can see that in all such examples whereX(R) → Y (R) is complicated, X(R) contains a special surface of nonnegative Eulercharacteristic. This turns out to be a general pattern, though the proof presentedhere relies on a laborious case analysis. The precise technical theorem is stated in(1.8).

None of the complicated examples occur if X(R) is orientable, and this yieldsthe following:

Theorem 1.2. Let X be a smooth, projective, real algebraic 3-fold and X∗ theresult of the MMP over R. Assume that X(R) is orientable.

Then the topological normalization (cf. 1.5) X∗(R) → X∗(R) is a PL-manifoldand X(R) can be obtained from X∗(R) by repeated application of the followingoperations:

(0) throwing away all isolated points of X∗(R),(1) taking connected sums of connected components,(2) taking the connected sum with S1 × S2,(3) taking the connected sum with RP3.

Remark 1.3. X∗ uniquely determines (1.2.0) and also (1.2.1). The latter can beseen by analyzing real analytic morphisms h : [0, 1] → X∗(R) where the endpointsmap to different connected components of X∗(R). In practice this may be quitehard, and it could be easier to work through the MMP backwards.

X∗ contains some information about the steps (1.2.2–3), but these are by nomeans unique. Even if X∗ is smooth, both of these steps are possible, as shown bythe next example.

Example 1.4. It is well known how to create the connected sum with RP3 al-gebraically. Let X be a smooth 3-fold over R and 0 ∈ X(R) a real point. SetY = B0X . Then Y (R) ∼ X(R) # RP3. (X(R) and RP3 are orientable but not ori-ented. The connected sum of two nonoriented manifolds is, in general, not unique.It is, however, unique if one of the summands has an automorphism with an isolatedfixed point which reverses local orientation there.)

The connected sum with S1×S2 is somewhat harder. Let X be a smooth 3-foldover R and D ⊂ X a real curve which has a unique real point {0} = D(R). Assumefurthermore that near 0 the curve is given by equations (z = x2 + y2 = 0). SetY1 = BDX . Y1 has a unique singular point P ; set Y = BP Y1. It is not hard to seethat Y is smooth and Y (R) ∼ X(R) # (S1 × S2).

1.5 (Piecewise linear 3–manifolds). As (1.2) already shows, we have to move be-tween topological, PL and differentiable manifolds. In this paper we usually workwith piecewise linear manifolds (see [Rourke-Sanderson82] for an introduction).


36 JANOS KOLLAR

In dimension 3 every compact topological 3–manifold carries a unique PL–manifold structure (cf. [Moise77, Sec. 36]) and also a unique differentiable struc-ture (cf. [Hempel76, p.3]). We mostly use the PL–structure since most algebraicconstructions are natural in the PL–category. For instance, R1 → R2 given byt 7→ (t2, t3) can be viewed as a PL–embedding but not as a differentiable embed-ding in the natural differentiable structures.

In dimension 3 the PL–structure behaves very much like a differentiable struc-ture. For instance, let M3 be a PL 3–manifold, N a compact PL–manifold ofdimension 1 or 2 and g : N ↪→ M a PL–embedding. Then a suitable open neigh-borhood of g(N) is PL–homeomorphic to a real vector bundle over N (cf. [Moise77,Secs. 24 and 26]). (Note that a similar result fails for topological 3–manifolds (cf.[Moise77, Sec. 18]), and it also fails for PL 4–manifolds: take any nontrivial knotin S3 and suspend it in S4.)

The technical definition of “suitable” neighborhoods is given by the notion ofregular neighborhoods; see [Rourke-Sanderson82, Chap.3]. If f : M → N is a PL–map and X ⊂ N a compact subcomplex, then there is a regular neighborhoodX ⊂ U ⊂ N such that f−1(U) is a regular neighborhood of f−1(X) ⊂ M (cf.[Rourke-Sanderson82, 2.14]).

Let F be a complex with only finitely many points P ∈ F such that F is not aPL–manifold at P . I define the topological normalization p : F → F as the uniqueproper PL–morphism such that p−1 exists and is a homeomorphism at points whereF is a manifold and p−1(P ) is in a one–to–one correspondence with the connectedcomponents of the punctured neighborhood of P ∈ F otherwise. (This definitionclearly comes from algebraic geometry. It should not be confused with the unrelatednotion of “normal topological space”.)

1.6 (Surfaces in 3–manifolds). Let M be a PL 3–manifold without boundary, N acompact PL 2–manifold without boundary and g : N ↪→ M a PL–embedding. Aswe noted above, a neighborhood of N is an R-bundle over N . R-bundles over Nare classified by group homomorphisms ρ : π1(N) → {±1}. If ρ is trivial, then Nis 2–sided in M , otherwise it is 1–sided . We also allow self-homeomorphisms of N ,thus we get the following possibilities when N has nonnegative Euler characteristic:

S2: Always 2–sided, many such surfaces in every M3.RP2: M3 is not orientable in the 2–sided case. Such manifolds are called P2-

reducible (cf. [Hempel76, p.88]). In the 1–sided case the boundary of aregular neighborhood is S2, thus M ∼ M ′ # RP3 for some 3–manifold M ′.Most 3–manifolds do not contain any RP2.

Torus: The 2–sided case occurs in any 3–manifold as the boundary of a regularneighborhood of any S1 along which M is orientable. There is a unique 1–sided case. For these M is not orientable. Most nonorientable 3–manifoldsdo not contain 1–sided tori; see section 12.

Klein bottle: M is nonorientable in the 2–sided case. The boundary of aregular neighborhood of any S1 along which M is nonorientable is such.There are two different 1–sided cases, depending on whether M is orientablenear N or not. These are again rare; see section 12.

This shows that there are many 3–manifolds which do not contain RP2, 1–sidedtori or Klein bottles. These correspond to 6 different cases on the above list. Itturns out that we need to exclude only 3 of these for our main theorem.



Condition 1.7. Let M be a PL 3-manifold without boundary. Consider the fol-lowing properties:

(1) M does not contain a 2-sided RP2;(2) M does not contain a 1-sided torus;(3) M does not contain a 1-sided Klein bottle with nonorientable neighborhood.

Failure of any of these properties implies that M is not orientable, but there aremany nonorientable 3-manifolds which do satisfy all 3 of the above conditions. Forinstance, this holds if M is hyperbolic (12.1).

Theorem 1.8. Let X be a smooth, projective, real algebraic 3-fold and X∗ theresult of the MMP over R. Assume that X(R) satisfies the 3 conditions (1.7.1–3).

Then the conclusions of (1.2) hold.

Remark 1.9. It would seem that we also need to allow the connected sum withS1×S2 (cf. (5.1)), corresponding to attaching a nonorientable 1–handle. This,however, would give a 1–sided torus which we excluded.

All 3 conditions (1.7.1–3) are necessary for the theorem to hold. My feeling isthat essentially nothing can be said without (1.7.1) or (1.7.3). (1.7.2) has a twofoldrole in the proof. First, it ensures that X is not obtained as a blow up of a smooth3-fold Y along a curve. This in itself would not be a problem, but it may happenthat Y (R) contains a 2-sided RP2 but X(R) does not. It seems to me that thisleads to rather complicated topological questions. Still, a suitable reformulation ofthe theorem may get around this problem. Second, (1.7.2) is also used to excludea few singularities on the Xi. These cases are of index 1 (3.1) and they can bedescribed very explicitly. It should be possible to work with them.

The technical heart of the proof is a listing of the possible singularities thatoccur in the course of the MMP and a fairly detailed description of the steps ofthe MMP. The final result is relatively easy to state but the proof is a case-by-caseexamination.

Theorem 1.10. Let X be a smooth, projective, real algebraic 3-fold and assumethat X(R) satisfies the 3 conditions (1.7.1–3).

Let Xi be any of the intermediate steps of the MMP over R starting with X, andlet 0 ∈ Xi(R) be a real point. Then a neighborhood of 0 ∈ Xi is real analyticallyequivalent to one of the following standard forms:

(1) (cA0) Smooth point.(2) (cA+

>0) (x2 + y2 + g≥2(z, t) = 0), where g is not everywhere negative in apunctured neighborhood of 0.

(3) (cE6) (x2 + y3 + (z2 + t2)2 + yg≥4(z, t) + g≥6(z, t) = 0).

Remark 1.11. The symbol g≥m denotes a power series of multiplicity at least m.The name of the cases is explained in [Kollar97b].The above points of type cE6 form a codimension 6 family in the space of all cE6

singularities. They all occur, even if X(R) is orientable. Points of type cA+>0 occur

for many choices of g. Section 10 gives an algorithm to decide which cases of g dooccur, but I was unable to write the condition in closed form. For the applicationsthis does not seem to matter.

Using [Kollar97b, 4.3, 4.4, 4.9], this immediately implies:

Corollary 1.12. With the notation and assumptions given above, we have thatXi(R) \ {isolated points} is a compact PL 3-manifold without boundary.


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The next step is to understand the “elementary” steps of the MMP over R. (1.8)turns out to be a consequence of (1.13). (See (9.1) for the definition of weightedblow-ups.)

Theorem 1.13. Let X be a smooth, projective, real algebraic 3-fold such that X(R)satisfies the conditions (1.7.1–3).

Let fi : Xi 99K Xi+1 be any of the intermediate steps of the MMP over R startingwith X. Then the induced map fi : Xi(R) → Xi+1(R) is everywhere defined andthe following is a complete list of possibilities for fi:

(1) (R-trivial) fi is an isomorphism in a (Zariski) neighborhood of the set ofreal points.

(2) (R-small) fi : Xi(R) → Xi+1(R) collapses a 1-complex to points and thereare small perturbations fi of fi such that fi : Xi(R) → Xi+1(R) is a PL-homeomorphism.

(3) (smooth point blow up) fi is the inverse of the blow up of a smooth pointP ∈ Xi+1(R).

(4) (singular point blow up) fi is the inverse of a (weighted) blow up of a singularpoint P ∈ Xi+1(R). There are two cases:(a) (cA+

>0, mult0 g even) Up to real analytic equivalence near P , Xi+1∼=

(x2 + y2 + g≥2m(z, t) = 0) where g2m(z, t) 6= 0 and Xi is the weightedblow up B(m,m,1,1)Xi+1.

(b) (cA+>0, mult0 g odd) Up to real analytic equivalence near P , Xi+1

∼=(x2 + y2 + g≥2m+1(z, t) = 0) where z2m+1 ∈ g and zitj 6∈ g for 2i + j <4m + 2. Xi is the weighted blow up B(2m+1,2m+1,2,1)Xi+1.

For inductive purposes the following generalization of (1.13) is frequently useful:

Theorem 1.14. Let X be a projective real algebraic variety of dimension 3. As-sume that the following 3 conditions hold:

(1) X has terminal singularities only (3.3),(2) KX is Cartier along X(R),(3) X(R) is a PL 3–manifold which satisfies the conditions (1.7.1–3).

Then every intermediate step fi : Xi 99K Xi+1 of the real MMP starting with X isone of those listed in (1.13) and each Xi+1 satisfies the above 3 conditions.

The R-trivial steps do not change anything in a neighborhood of the real points,but it is in these steps that the full complexity of the MMP appears. All thedifficulties involving higher index terminal singularities (3.1, 3.3) and flips (3.10)are present, but they always appear in conjugate pairs with no real points.

For the topological questions these have no effect, but in other applications of(1.13) this should be taken into account.

Remark 1.15. The lists in (1.10) and (1.13) are fairly short, but I do not see asimple conceptual way of stating the results, let alone proving them by generalarguments. The appearence of the singularities of type cE6 in (1.10) was ratherunexpected for me.

The formulations also hide the circumstance that there does not seem to be asingle method of excluding all other a priori possible cases. The algebraic methodof the proof of (1.10) ends with a much longer list (8.2). The topological methodexcludes many of these right away, but in a few cases several steps of the MMPneed to be analyzed.



1.16 (Method of the proof of (1.13)). The proof relies on rather extensive com-putations. The first step is a classification of all 3–dimensional terminal singular-ities over R and the study of their topological properties. This was carried out in[Kollar97b]. The next step is to gain a good understanding of the resolutions ofthese singularities. More precisely, we need to understand the “simplest” excep-tional divisors in these resolutions. (Simplicity is measured by the discrepancy, cf.(3.3).) Over C the first step in this direction is [Markushevich96]. A much moredetailed study of such exceptional divisors was completed by [Hayakawa97]. Ourmain emphasis is over R, and it turns out that there is very little overlap betweenthe computations of [Hayakawa97] and those in sections 9–11. Nonetheless, thebasic underlying principles are exactly the same.

2. Applications and speculations

2.1. Factorization of birational morphisms. Let f : Y → X be a birationalmorphism between smooth and projective varieties. It is a very old problem tofactor f as a composite of “elementary” birational morphisms. In dimension 2 thisis easy to do: f is the composite of blow ups of points. In dimension 3 and over C,the MMP factors f as a composition of divisorial contractions and flips (3.10), butthese intermediate steps are rather complicated and not too well understood.

If f : Y → X is a birational morphism between smooth and projective threefoldsover R, then one would like to get a factorization where the intermediate steps arealso defined over R. It turns out that if Y (R) is orientable, the answer is verysimple. As with minimal models in general, the intermediate steps involve singularvarieties, though in this case the real singularites are very mild.

Definition 2.1. A real 3–fold X is said to have a cA1 singularity at 0 ∈ X(R) ifin suitable real analytic coordinates X can be given by an equation (±x2 ± y2 ±z2 ± tm = 0) for a suitable choice of signs and m ≥ 1.

Theorem 2.2. Let f : Y → X be a birational morphism between smooth andprojective threefolds over R. Assume that Y (R) satisfies the conditions (1.7). Thenf can be factored as

f : Y = Xnfn→ Xn−1 → · · · → X1

f1→ X0 = X,

where each Xi has only cA1 singularities at real points and the following is a com-plete list of possibilities for the fi:

(1) (smooth point blow up) fi is the blow up of a smooth point P ∈ Xi−1(R).(2) (singular point blow up) fi is the blow up of a singular point P ∈ Xi−1(R).(3) (curve blow up) fi is the blow up of a real curve C ⊂ Xi−1. C has only

finitely many real points, Xi−1 is smooth at each of these and in suitable realanalytic coordinates C can be written as (z = x2 + y2m = 0).


Remark 2.3. As in (1.13), it is in the R-trivial steps that the full complexity ofthe MMP appears. In particular, the R-trivial steps may be flips (3.10) where theflipping curve has no real points.

Proof. For purposes of induction we consider the more general case when X isallowed to have cA1-type singularities at real points and terminal (3.3) singularities


40 JANOS KOLLAR

at complex points. We assume that X is Q-factorial (that is, a suitable multiple ofevery Weil divisor is Cartier). Run the real MMP for Y over X to obtain

f : Y = Xnfn99K Xn−1 99K · · · 99K X1

f1→ X0 = X.

The proof is by induction on the number of steps it takes the MMP to reach X .The last step, f1 : X1 → X0 = X , is a contraction since we work over X . The

possibilities for f1 are described in (1.13). We are done by induction if f1 is R-trivialor a smooth point blow up. Assume that f1 is a singular point blow up. Since X0

has only cA1 points, we are in case (1.13.4a) with m = 1. f1 is the ordinary blowup and by explicit computation we see that X1 still has only cA1 singularities.

The case when f1 is R-small (1.13.2) needs to be studied in greater detail. f1

cannot be a g–extraction (6.6) since cA1 type points do not have g–extractionsother than the blow up above by (8.2). Thus f1 is the blow up of a curve C ⊂ X0.Moreover, X0 is smooth along C(R) and C is locally planar along C(R) by (7.2).C(R) is finite since f1 is R-small.

Pick any point P ∈ C(R) and assume that C is given by real analytic equations(z = g(x, y) = 0). By explicit computation, BCX0 has a unique singular point withequation (st− g(x, y) = 0) which is equivalent to (u2 − v2 − g(x, y) = 0).

X1 is an intermediate step of an MMP starting with Y , hence its singularitiesare among those listed (1.10). Thus g has multiplicity 2 and so it can be written as±x2±yr. Since (g = 0) has only the origin as its real solution, g = ±(x2+y2m).

2.2. Application to the Nash conjecture. The main conclusion of (1.2) and(1.8) is that if we want to understand the topology of X(R) (say when it is ori-entable), it is sufficient to study the topology of X∗(R) instead. X∗ has varioususeful properties, depending on the conditions imposed on X .

Consider, for instance, the original Nash question: What happens if X is rational.Since the fifties it has been understood that being rational is a very subtle conditionand it is very hard to work with. [KoMiMo92] introduced the much more generalnotion of being rationally connected . X is rationally connected iff two general pointsof X(C) can be connected by an irreducible rational curve. The lines show that Pn

is rationally connected.The structure theory of [KoMiMo92] implies that a 3-fold X is rationally con-

nected iff X∗ falls in one of 3 classes:

(1) (Conic fibrations) There is a morphism (over R) g : X∗ → S onto a rationalsurface such that the general fiber is a conic. Correspondingly there is amorphism X∗(R) → S(R) whose general fiber is S1 or empty. These casesare studied in [Kollar98a].

(2) (Del Pezzo fibrations) There is a morphism (over R) g : X∗ → C onto arational curve such that the general fiber is a Del Pezzo surface. If X(R)is orientable, then this induces a morphism X∗(R) → C(R) whose generalfiber is a torus or a union of some copies of S2. These cases are studied in[Kollar98b].

(3) (Fano varieties) The anticanonical bundle of X∗ is ample. There is a com-plete list of such varieties if X∗ is also smooth [Iskovskikh80]. Even if X∗ isknown rather explicitly, a topological description of X∗(R) may not be easy.It would be interesting to work out at least some of the cases, for instancehypersurfaces of degree 3 or 4 in P4. (Mikhalkin pointed out that the degree



3 cases can be understood using the classification of degree 4 real surfacesin RP3 [Kharlamov76].)

In general it is known that there are only finitely many families of singularFano varieties in dimension 3 [Kawamata92]. Thus we can get only finitelymany different topological types for X∗(R) in this case.

The precise version of the above results is summarized in [Kollar98c].

2.3. Homology spheres. It is interesting to consider if we can get additionalsimplifications of the real MMP if we pose further restrictions on X(R). We mayassume, for instance, that X(R) is a homology sphere. This was in fact the as-sumption considered first. One can ask if under this assumption X(R) → X∗(R) isa homeomorphism.

Unfortunately this is not the case. Consider for instance the singular real three-fold X∗ given by the affine equation

x2 + y2 + z2 + (t− a0)(t− am)m−1∏i=1

(t− ai)2r = 0,

where a0 < a1 < · · · < am are reals. This has m − 1 singular points of the formx2 + y2 + z2 − u2r = 0, which can be resolved by r successive blow ups. Resolvingall singular points we obtain the 3–fold X . One can easily see that X(R) ∼ S3, butX∗(R) is the disjoint union of m copies of S3.

One may also study the types of singularities that occur if we pose strongerrestrictions on X(R). It seems that the best one can get is the following:

Conjecture 2.4. Let X be a smooth projective 3–fold over R. Assume that X(R)satisfies the conditions (1.7) and X(R) cannot be written as a connected sum withS1 × S2.

Let Xi be any of the intermediate steps of the MMP over R starting with X, andlet 0 ∈ Xi(R) be a real point. Then a neighborhood of 0 ∈ Xi is real analyticallyequivalent to one of the following standard forms:

(1) (cA0) Smooth point.(2) (cA+

>0) (x2+y2+g≥2(z, t) = 0), where mult0 g is even and g is not everywherenegative in a punctured neighborhood of 0.

In fact, most cA+>0-type singularities should not occur. It is possible that one can

write down a complete list. Also, one can be more precise about how the singularpoints separate Xi(R).

The results in sections 8–11 come close to proving (2.4), but two points remainunresolved. In order to exclude cE6 type points, one needs to show that the onlypossible g–extraction is the one described in (11.6). This should be a feasiblecomputation. The main problem is that in (10.10) I could not exclude certainR-small contractions. I do not see how to deal with this case.

2.4. The relative case. Let X be a smooth real algebraic threefold and g : X → Sa projective morphism defined over R. g may be birational, but it is also of interestto study the cases when X is written as a family of curves or surfaces. In thesecases it is natural to consider a relative MMP where all the intermediate steps Xi

are required to admit compatible morphisms gi : Xi → S.The results of the introduction all apply in this setting as well. There are,

however, some cases when it is possible to drop all topological assumptions about


42 JANOS KOLLAR

X(R). We do not even need to assume that X(R) is compact. This happens, forinstance, when the real fibers of g are curves. (g is allowed to have 2-dimensionalcomplex fibers but each fiber has only a 1-dimensional set of real points.) Thedivisor to point contractions discussed in (8.2) cannot happen, but we can have theinverses of blow ups of curves classified in (7.2). Thus we obtain:

Theorem 2.5. Let X be a real algebraic variety of dimension 3 and g : X → S aprojective morphism defined over R. Assume that the following 4 conditions hold:

(1) X has terminal singularities only,(2) KX is Cartier along X(R),(3) X(R) is a PL 3–manifold,(4) every fiber of g : X(R) → S(R) has (real) dimension at most 1.

Let fi : Xi 99K Xi+1 by any of the intermediate steps of the real relative MMP overS starting with X. Then Xi+1 satisfies the above conditions (1–3) and fi is one ofthe following:


(6) (R-small) fi : Xi(R) → Xi+1(R) collapses a 1-complex to points and thereare small perturbations fi of fi such that fi : Xi(R) → Xi+1(R) is a PL-homeomorphism.

(7) (curve blow up) fi is the inverse of the blow up of a curve C ⊂ Xi+1. C(R)is a PL manifold, Xi+1 is smooth along C(R) and C is locally planar alongC(R).

2.5. Real closed fields. It is sometimes important to understand real algebraicvarieties when the base field R is replaced by an arbitrary real closed field R. Thetheorems of this paper can be generalized to this setting as follows.

There is no natural Euclidean topology on the set X(R), but it is possible todefine homology and cohomology groups such that they coincide with the usual onesfor R = R (cf. [BCR87, Sec.11.7]). Moreover, this definition works for semialgebraicsets.

A compact 2–manifold without boundary is determined by its homology. Theorientability of neighborhoods and 1 or 2–sidedness is again a homological question.Thus it is reasonable to replace (1.7) with the following:

(∗) X(R) does not contain any semialgebraic subset S such that S and its neigh-borhood have the same homologies as those excluded in (1.7.1–3).

With this modification, the rest of the proof goes through with essentially nochanges.

The conditions (1.7.1–3) are used mainly in (8.4). There we use (1.7) only forcomponents of an algebraic subset. A slight complication arises in the inductivesteps. If we find a surface F ⊂ Xi such that one of the components of F (R) violates(1.7), then we would like to claim that we can find a similar surface in the originalvariety X . X is obtained from Xi typically by blowing up points. In Xi(R) a surfacecan be moved away from finitely many points, as long as Xi(R) is a manifold.

Let Z be any smooth variety over a real closed field and S ⊂ Z a smoothsurface. Then S can be moved away from finitely many points without changing itshomology groups. Unfortunately, we have to deal with some cases where S and Zare both singular. In such cases it may be impossible to move S away from singularpoints of Z. (For instance, let X be the cone over RP2 with vertex P and S the



cone over a line L ⊂ RP2. Then S is a disc but S cannot be embedded into Z \ Psuch that its boundary goes to L since L ⊂ Z \ P is not contractible.)

We have to use the special nature of our singularities. We know that Xi(R)is locally a cone over a disjoint union of copies of S2. Let p ∈ Xi(R) be a pointthat we want to move away from and L ⊂ Xi(R) the bundary of a small ball Baround p. If S is smooth, then S∩L is a circle which bounds a disc D in one of thecopies of S2. Remove S ∩B from S and paste D to its place to obtain a surface S′.Clearly S′ is homeomorphic to S and it does not pass through P . Moreover, S′ isa semialgebraic subset of Xi(R).

This operation makes sense over any real closed field and with its repeated usethe rest of (8.6) works.

This raises the question of how easy it is to check (∗). For instance, let X bea real algebraic 4–fold mapping to a curve C and R a real closure of R(C). Onecan expect that XR satisfies (∗) iff almost all fibers of F over C(R) satisfy (1.7).The question makes sense over higher dimensional bases as well. I do not know theanswer.

2.6. Beyond the Nash conjecture. One can refine the 3–dimensional Nash con-jecture in two ways.

One can study the topology of X(R) for other classes of real algebraic varieties.The simplest cases may be those whose minimal models admit a natural fibration.This should be very helpful in their topological study. One such class is ellipticthreefolds, where we have a morphism X∗ → S whose general fiber is an ellipticcurve. A study of the singular fibers occurring in codimension 1 was completed by[Silhol84].

Another, probably more difficult class is Calabi–Yau 3–folds. It would be veryinteresting to find some connection between the topology of X(R) and mirror sym-metry.

The following question is consistent with the known examples:

Question 2.6. Let X be a smooth projective real 3–fold. Assume that X(R) ishyperbolic. Does this imply that X is of general type?

One can also start with a 3–manifold M and look for a “simple” real projective3–fold X such that X(R) ∼ M . Ideally one would like to find a solution wherecertain topological structures on M are reflected by the algebraic properties of X .

There are hyperbolic 3–manifolds which embed into R4. This implies that theycan be realized by real algebraic hypersurfaces in R4. It would be interesting tofind such examples.

The methods of this paper require a very detailed study of the steps of the MMP,which is currently feasible only in dimension 3. It would be, however, interestingto develop some examples in higher dimensions.

Example (1.4) describing connected sum with S1 × S2 should have interestinghigher dimensional versions. There may be other, more complicated examples aswell.

The first steps of the 4–dimensional MMP over C have been recently classifiedby [Andreatta-Wisniewski96]. It should be possible to obtain the complete list overR and to study their topology.


44 JANOS KOLLAR

3. The minimal model program over R

This section is intended to provide a summary of the MMP over R. More gener-ally, we discuss the MMP over an arbitrary field K of characteristic zero, since thereis no difference in the general features. Conjecturally the whole program works inall dimensions but at the moment it is only established in dimensions ≤ 3.

[Kollar87, Kollar90] provide general introductions. The minimal model programfor real algebraic surfaces is explained in detail in [Kollar97a]. For more compre-hensive treatments (mostly over C) see [CKM88, Kollar et al.92, Kollar-Mori98].

One of the special features of the 3-dimensional MMP is that we have to workwith certain singular varieties in the course of the program.

Definition 3.1. Let X be a normal variety defined over a field K. A (Weil) divisorover K is a formal linear combination D :=

∑aiDi (ai ∈ Z) of codimension 1

subvarieties, each defined and irreducible over K. A Q-divisor is defined similarly,except we allow ai ∈ Q. A divisor D is called Cartier if it is locally definable byone equation and Q-Cartier if mD is Cartier for some m 6= 0. The smallest suchm > 0 is called the index of D.

We say that X is factorial (resp. Q-factorial) if every Weil divisor is Cartier(resp. Q-Cartier).

A divisor D defined over K is Cartier (resp. Q-Cartier) iff it is Cartier (resp.Q-Cartier) after some field extension. However, a variety may be Q-factorial overK and not Q-factorial over K. For instance, the cone x2 + y2 + z2 − t2 is factorialover R but not over C. (For instance, (x−√−1y = z − t = 0) is not Q-Cartier.)

Definition 3.2. For a normal variety X , let KX denote its canonical class . KX

is a linear equivalence class of Weil divisors. The corresponding reflexive sheafOX(KX) is isomorphic to the dualizing sheaf ωX of X .

The index of KX is called the index of X .

Definition 3.3. Let X, Y be normal varieties and f : Y → X a birational mor-phism with exceptional set Ex(f). Let Ei ⊂ Ex(f) be the exceptional divisors. IfmKX is Cartier, then f∗OX(mKX) is defined and there is a natural isomorphism

f∗OX(mKX)|(Y \ Ex(f)) ∼= OY (mKY )|(Y \ Ex(f)).

Hence there are integers bi such that

OY (mKY ) ∼= f∗OX(mKX)(∑

biEi).

Formally divide by m and write this as

KY ≡ f∗(KX) +∑

a(Ei, X)Ei, where a(Ei, X) ∈ Q.

The rational number a(Ei, X) is called the discrepancy of Ei with respect to X .The closure of f(Ei) ⊂ X is called the center of Ei on X . It is denoted by

centerX Ei.If f ′ : Y ′ → X is another birational morphism and E′

i := (f ′ ◦ f−1)(Ei) ⊂ Y ′

is a divisor, then a(E′i, X) = a(Ei, X) and centerX Ei = centerX E′

i. Thus thediscrepancy and the center depend only on the divisor up to birational equivalence,but not on the particular variety where the divisor appears.

Definition 3.4. Let X be a normal variety such that KX is Q-Cartier. We saythat X is terminal (or that it has terminal singularities) if for every f : Y → X ,the discrepancy of every exceptional divisor is positive.



The following result makes it feasible to decide if X is terminal or not (cf.[CKM88, 6.5] or [Kollar-Mori98, 2.32]).

Lemma 3.5. For a normal variety X the following are equivalent:

(1) X is terminal.(2) a(E, X) > 0 for every resolution of singularities f : Y → X and for every

exceptional divisor E ⊂ Ex(f).(3) There is a resolution of singularities f : Y → X such that a(E, X) > 0 for

every exceptional divisor E ⊂ Ex(f).

Example 3.6. It is frequently not too hard to compute discrepancies. Assume forinstance that X is a hypersurface defined by (F (x1, . . . , xn) = 0). A local generatorof OX(KX) is given by any of the forms

ηi :=1

∂F/∂xidx1 ∧ · · · ∧ dxi−1 ∧ dxi+1 ∧ · · · ∧ dxn.

Let f : Y → X be a resolution of singularities and P ∈ Y a point with localcoordinates y1, . . . , yn−1. f is given by coordinate functions xi = fi(y1, . . . , yn−1)and so we can write

f∗ηn = f∗(

1∂F/∂xn

)Jac dy1 ∧ · · · ∧ dyn−1, where

Jac = Jac(

f1, . . . , fn−1

x1, . . . , xn−1

)denotes the determinant of the Jacobian matrix. Hence the discrepancies can becomputed as the order of vanishing of the Jacobian minus the order of vanishing off∗(∂F/∂xn).

If X is smooth, then we conclude that a(E, X) ≥ 1 for every exceptional divisor.Thus smooth varieties are terminal.

Next we define various birational maps which have a special role in the MMP.

Definition 3.7. Let X be a variety over K and assume that KX is Q-Cartier.A proper morphism g : X → Y is called an extremal contraction if the followingconditions hold:

(1) g∗OX = OY .(2) X is Q-factorial.(3) Let C ⊂ X be any irreducible curve such that g(C) = point. Then a

Q-divisor D on X is the pull back of a Q-Cartier Q-divisor D′ on Y iff(D · C) = 0. (Necessarily, D′ = g∗(D).)

Definition 3.8. Let g : X → Y be an extremal contraction.We say that g is of fiber type if dimY < dim X .We say that g is a divisorial contraction if the exceptional set Ex(g) is the

support of a Q-Cartier divisor. In this case Ex(g) is irreducible over K.We say that g is a small contraction if dim Ex(g) ≤ dim X − 2.One can see that every extremal contraction is in one of these 3 groups.

Definition 3.9. A proper morphism f : X → Y is called KX -negative if −KX isf -ample.


46 JANOS KOLLAR

Definition 3.10. Let f : X → Y be a small KX-negative extremal contraction. Avariety X+ together with a proper birational morphism f+ : X+ → Y is called aflip of f if

(1) KX+ is Q-Cartier,(2) KX+ is f+-ample, and(3) the exceptional set Ex(f+) has codimension at least two in X+.

By a slight abuse of terminology, the rational map φ : X 99K X+ is also called aflip. A flip gives the following diagram:

Xφ99K X+

−KX is f -ample ↘ ↙ KX+ is f+-ampleY

It is not hard to see that a flip is unique and the main question is its existence.

We are ready to state the 3-dimensional MMP over an arbitrary field:

Theorem 3.11 (MMP over K). Let X be a smooth projective 3-fold defined overa field K (of characteristic zero). Then there is a sequence

X = X0f099K X1 99K · · · 99K Xi

fi99K Xi+1 99K · · · fn−199K Xn =: X∗

with the following properties:(1) Each Xi is a terminal projective 3-fold over K which is Q-factorial over K.(2) Each fi is either a KX-negative divisorial extremal contraction or the flip of

a KX-negative small extremal contraction.(3) One of the following holds for X∗:

(a) KX∗ is nef (that is, (C ·KX∗) ≥ 0 for any curve C ⊂ X∗), or(b) there is a fiber type extremal contraction X∗ → Z.

Remark 3.12. For the purposes of this paper one can handle the MMP as a blackbox. It is sufficient to know that it works, but we will use very few of its finerproperties. In particular, there is no need to know anything about flips beyondbelieving their existence.

The rest of the section is devoted to explicitly stating all further results fromminimal model theory that are used later. The most significant among these isthe classification of terminal 3-fold singularities over nonclosed fields, establishedin [Kollar97b].

Notation 3.13. For a field K let K[[x1, . . . , xn]] denote the ring of formal powerseries in n variables over K. For K = R or K = C, let K{x1, . . . , xn} denotethe ring of those formal power series which converge in some neighborhood of theorigin.

For a power series F , Fd denotes the degree d homogeneous part. The multiplic-ity, denoted by mult0 F , is the smallest d such that Fd 6= 0. If we write a powerseries as F≥d, then it is assumed that its multiplicity is at least d.

For F ∈ R{x1, . . . , xn} let (F = 0) denote the germ of its zero set in Cn withits natural real structure. We always think of it as a complex analytic germ with areal structure and not just as a real analytic germ in Rn.

(F = 0)/ 1n (a, b, c, d) means the following. Define a Zn-grading of C{x, y, z, t} by

x 7→ a, y 7→ b, z 7→ c, t 7→ d. If F is graded homogeneous, then (F = 0)/ 1n (a, b, c, d)



denotes the germ whose ring of holomorphic functions is the ring of grade zeroelements of C{x, y, z, t}/(F ).

If (F = 0) is terminal, then n coincides with the index (3.2) of the singularity.

Example 3.14. In case X = (x2 + y2 + z2 + t2 = 0)/ 12 (1, 1, 1, 0) the ring is

OX = C{x2, y2, z2, t, xy, yz, zx}/(x2 + y2 + z2 + t2),

with the natural real structure.X can also be realized as the image of the hypersurface (x2 + y2 + z2 + t2 = 0)

under the map

φ : C4 → C7 : (x, y, z, t) 7→ (x2, y2, z2, t, xy, yz, zx),

which has degree 2 over its image.Although (x2 + y2 + z2 + t2 = 0) has only the origin as its real solution, X has

plenty of real points. Indeed, any real solution of x2 + y2 + z2 − t2 = 0 gives a realpoint P = φ(

√−1x,√−1y,

√−1z, t) ∈ X(R). φ−1(P ) is a pair of conjugate pointson the hypersurface (x2 + y2 + z2 + t2 = 0). All the real elements of OX take upreal values at P .

This way we see that X(R) is a cone over 2 copies of RP2.

The following is a summary of the classification of terminal singularities obtainedin [Kollar97b]. As it turns out, the classification closely follows the earlier resultsover algebraically closed fields. The choice of the subdivison into cases is dictatedby the needs of the proof in sections 9–11, rather than the internal logic of theclassification.

Theorem 3.15. Let X be a real algebraic or analytic 3-fold and 0 ∈ X(R) a realpoint. Then X has a terminal singularity at 0 iff a neighborhood of 0 ∈ X is realanalytically equivalent to one of the following:

name equationcA0 (t = 0)cA1 (x2 + y2 ± z2 ± tm = 0)cA+

>1 (x2 + y2 + g≥3(z, t) = 0)cA−

>1 (x2 − y2 + g≥3(z, t) = 0)cD4 (x2 + f≥3(y, z, t) = 0), where f3 6= l21l2 for linear forms licD>4 (x2 + y2z + f≥4(y, z, t) = 0)cE6 (x2 + y3 + yg≥3(z, t) + h≥4(z, t) = 0), where h4 6= 0cE7 (x2 + y3 + yg≥3(z, t) + h≥5(z, t) = 0), where g3 6= 0cE8 (x2 + y3 + yg≥4(z, t) + h≥5(z, t) = 0), where h5 6= 0cA0/n (t = 0)/ 1

n (r,−r, 1, 0), where n ≥ 2 and (n, r) = 1cA1/2 (x2 + y2 ± zn ± tm = 0)/ 1

2 (1, 1, 1, 0), where min{n, m} = 2cA+

>1/2 (x2 + y2 + f≥3(z, t) = 0)/ 12 (1, 1, 1, 0)

cA−>1/2 (x2 − y2 + f≥3(z, t) = 0)/ 1

2 (1, 1, 1, 0)cA/n (xy + f(z, t) = 0)/ 1

n (r,−r, 1, 0), where n ≥ 3 and (n, r) = 1cAx/2 (x2 ± y2 + f≥4(z, t) = 0)/ 1

2 (0, 1, 1, 1)cAx/4 (x2 ± y2 + f≥2(z, t) = 0)/ 1

4 (1, 3, 1, 2)cD/2 (x2 + f≥3(y, z, t) = 0)/ 1

2 (1, 1, 0, 1)cD/3 (x2 + f≥3(y, z, t) = 0)/ 1

3 (0, 1, 1, 2), where f3(0, 0, 1) 6= 0cE/2 (x2 + y3 + f≥4(y, z, t) = 0)/ 1

2 (1, 0, 1, 1)


48 JANOS KOLLAR

The list over any field of characteristic zero is similar, the only difference isthat each square term appears with an unknown coefficient. For instance cA+

>1

is the case (ax2 + by2 + g≥3(z, t) = 0) where −ab is not a square and cA−>1 is

(ax2 + by2 + g≥3(z, t) = 0) where −ab is a square.

4. The topology of real points and the MMP

Starting with a projective variety X over R, let us run the MMP over R. Weobtain a sequence of birational maps

X = X0 99K X1 99K · · · 99K Xifi99K Xi+1 99K · · · 99K X∗.

These in turn induce (not necessarily everywhere defined) maps between the setsof real points

X(R) = X0(R) 99K · · · 99K Xi(R)fi99K Xi+1(R) 99K · · · 99K X∗(R).

Our aim is to see if there is a way of describing X(R) in terms of X∗(R) and a localdescription of the maps Xi(R) 99K Xi+1(R) in a neighborhood of their exceptionalsets.

Proposition 4.1. Every step fi of the MMP over R is among the following five:

(1) (divisor–to–point) fi contracts a geometrically irreducible (1.1) divisor Ei ⊂Xi to a point Pi+1 ∈ Xi+1(R).

(2) (divisor–to–curve) fi contracts a geometrically irreducible divisor Ei ⊂ Xi

to a real curve Ci+1 ⊂ Xi+1.(3) (R-small) fi : Xi(R) → Xi+1(R) collapses a 1-complex to points and is a

homeomorphism elsewhere.(4) (flip) fi is the flip of a curve Ci ⊂ Xi.(5) (R-trivial) fi is an isomorphism in a (Zariski) neighborhood of the set of

real points.

Proof. If fi is a flip, then we have case (4). Thus we may assume that fi is thecontraction of a divisor Ei ⊂ Xi and Ei is irreducible over R. If Ei is irreducibleover C, then we have one of the cases (1–2). If Ei is reducible over C, then Ei(R)is a 1-complex by (4.2) and so we are in case (3).

Any of the above cases can also be of type (5).

Lemma 4.2. Let X be an n-dimensional scheme over R (that is, an algebraicvariety possibly with several irreducible components and with singularities). As-sume that if Xi ⊂ X is any R-irreducible component, then Xi is reducible overC. Then X(R) = (Sing X)(R), that is, every real point is singular. In particular,dim X(R) ≤ n− 1.

Proof. Assume that P ∈ X(R) is a smooth real point. Then P lies on a uniqueirreducible component Y ⊂ XC, thus Y is invariant under complex conjugation. SoY is an irreducible real component which stays irreducible over C, a contradiction.

Each of the 5 steps Xi(R) 99K Xi+1(R) have different topological behavior. Thefollowing informal discussion intends to emphasize their main features.



4.3 (Divisor–to–point). Let M = Xi(R) be a 3-complex (with only finitely manysingular points) and F = Ei(R) ⊂ M a 2-complex. We collapse F to a point:

F ⊂ M↓ ↓ fP ∈ N.

In practice we are frequently able to describe a regular neighborhood F ⊂ U ⊂ M(this is a local datum) and by assumption we know a regular neighborhood P ∈V ⊂ N . Thus we see that M is obtained from U and N \ IntV by gluing themtogether along the boundaries ∂U and ∂V .

The gluing is determined by a PL-homeomorphism φ : ∂U → ∂V . Thus, besidesknowing U and N , we also need to know φ up to PL-isotopy. If one of the connectedcomponents of ∂U has genus at least 2, this is a very hard problem. In fact, asthe example of Heegard splittings shows (cf. [Hempel76, Ch.2]), the choice of φ isusually the most significant information. Unfortunately, φ can be described only interms of global data.

If ∂U is a union of m copies of S2, then φ is classified by an element of thesymmetric group on m elements (which S2 maps where) and a sign for each S2

(describing whether the map is orientation preserving or reversing on that S2).Hence, knowing U and N , we can determine M up to finite ambiguity.

In many cases U is so simple that different choices of φ give the same M , givingeven fewer possibilities for M .

If P ∈ N is an isolated singular point, then ∂V is a union of spheres iff N (thetopological normalization of N) is a manifold.

The situation is similarly simple if ∂U is a union of copies of RP2 and of S2,and still manageable if ∂U also contains tori and Klein bottles. For us these moregeneral cases do not come up.

4.4 (Divisor–to–curve). This time we construct it bottom up. Assume for sim-plicity that N = Xi+1(R) is a 3-manifold and L = Ci+1(R) ⊂ N a link. Theprojectivized normal bundle is an S1-bundle S → L. The blow up of L in Nreplaces L by S to obtain:

S ⊂ M↓ ↓ fL ∈ N.

(In general N may have finitely many singular points and L is only a 1-complex,but I believe that a similar description is possible in all cases.)

Here M is uniquely determined, once we know N and L. By assumption we knowN but L is a free choice. The Jaco–Johannson–Shalen decomposition (cf. [Scott83,p.483]) shows that in most cases BLN determines M \ L. Thus the description ofall possible BLN is essentially equivalent to the description of all links.

For us L has to come from an algebraic curve, thus we are led to the question:Which links in a real algebraic 3-fold can be realized by algebraic curves? In somecases every link is realized (cf. [Akbulut-King81]), thus we again run into a hardtopological problem.

So M can be described in terms of N , though the answer depends on the choiceof a link, which is a very complicated object.


50 JANOS KOLLAR

4.5 (R-small contraction). N = Xi+1(R) is obtained from M = Xi(R) by collaps-ing a 1-complex C = (Sing Ei)(R) = Ei(R) to a point:

C ⊂ M↓ ↓ fP ∈ N.

If the normalizations M and N are manifolds, then we see in (5.3) that a suitablesmall perturbation of f is a homeomorphism between M and N . Thus this step(which is actually more complicated from the point of view of algebraic geometrythan the previous two cases) is easy to analyze topologically.

4.6 (Flip). Assume for simplicity that M = Xi(R) is an orientable 3-manifoldand C(R) ∼ S1. N = Xi+1(R) is obtained from M by a surgery along S1. Theboundary of a regular neighborhood of S1 is S1×S1, and the surgery is determinedby a diffeomorphism of S1×S1 up to isotopy. These are classified by SL(2, Z). (Ingeneral M may have finitely many singular points and C(R) is a 1-complex, but Ibelieve that a similar description is possible in all cases.)

A complete classification of flips is known [Kollar-Mori92], thus it should bepossible to compute the resulting diffeomorphism of S1 × S1.

Here again we run into a global problem. S1 ⊂ M may be knotted, and theresult of the surgery depends mostly on the knot S1 ⊂ M . The usual descriptionsof flips characterize a complex analytic neighborhood of C, thus they say nothingabout how its real part is knotted. From the point of view of algebraic geometry,this is a global invariant.

We have the additional problem that flipping curves are rigid objects, thus wecannot hope to get a flipping curve by approximating a real curve algebraically.Furthermore, it is very hard to determine which curves are obtained by a flip.(Even if Z is a smooth complex 3-fold and C ⊂ Z a smooth CP1, I know of nopractical way of determining if C ⊂ Z is obtained as a result of a flip.)

4.7 (Conclusion). Start with a projective 3-fold X over R and run the MMP overR:

X = X0 99K X1 99K · · · 99K Xifi99K Xi+1 99K · · · 99K X∗.

If we would like to understand the topology of X(R) in terms of X∗(R), then wehave to ensure that the MMP has the following properties:

(1) Xi(R) is a manifold for every i.(2) Each fi is either R-trivial or R-small or a divisor–to–point contraction.

(1.13) asserts that both of these conditions can be satisfied by imposing certainmild conditions on the topology of X(R).

5. The topology of divisorial contractions

The aim of this section is to describe some examples where the change of thetopology of a real algebraic variety under a divisorial contraction can be readilyunderstood by topological methods.

Notation 5.1. The disjoint union of two topological spaces is denoted by M ] N .Direct product is denoted by M ×N . The unique nontrivial S2-bundle over S1 isdenoted by S1×S2. This is obtained from [0, 1]× S2 by identifying the 2 ends viaan orientation reversing homeomorphism.

Homeomorphism of two topological spaces is denoted by M ∼ N .



We start with the study of R-small contractions:

Lemma 5.2. Let f : M → N be a proper PL-map between PL-manifolds of di-mension n ≥ 3. Assume that there is a 1-complex C ⊂ M and a finite set of pointsP ⊂ N such that f : M \ C → N \ P is a PL-homeomorphism.

Then M and N are PL-homeomorphic (by a small perturbation of f).

Proof. If C is collapsible to points, then a regular neighborhood of C in M is aunion of disjoint n-cells [Rourke-Sanderson82, 3.27] and we are done.

In order to see that C is collapsible to points, we may assume that P is a pointand N = Sn. Thus M is also a compact PL-manifold. M is orientable outsidethe codimension ≥ 2 subset C, hence it is orientable. Consider the exact homologysequences:

Hi(C) → Hi(M) → Hi(M, C) → Hi−1(C)↓ ↓ ↓ ↓

Hi(P ) → Hi(Sn) → Hi(Sn, P ) → Hi−1(P )

We compute Hi(M, C) = Hi(Sn, P ) from the second sequence and substitute intothe first to obtain that

H1(C) ∼= H1(M) and 0 = Hn−1(C) ∼= Hn−1(M).

By Poincare duality we conclude that H1(C) = 0, thus C is contractible.

Corollary 5.3. Let f : X → Y be a morphism of n-dimensional real algebraicvarieties, n ≥ 3. Assume that

(1) X(R) = M ] R and Y (R) = N ] R′ where M, N are PL-manifolds anddim R, dim R′ < n.

(2) f induces an isomorphism R ∼= R′.(3) Ex(f)(R) is a 1-complex.

Then X(R) is PL-homeomorphic to Y (R).

Proof. Set C = Ex(f)(R) and C ⊂ M ⊂ X(R) its preimage. Since C has dimension1, there is a one–to–one correspondence between the connected components of Cand the connected components of the boundary of a regular neighborhood of C.Hence f lifts to a morphism f : X(R) → Y (R). Thus (5.2) implies (5.3), and thehomeomorphism is given by a small perturbation of f .

Lemma 5.4. Let M be a 3–complex with only finitely many singular points. LetC ⊂ M be a compact 1–complex and C ⊂ IntU ⊂ M a regular neighborhood of C.Assume that one of the connected components of ∂U is RP2. Then there is a pointp ∈ C such that one of the connected components of the link of p ∈ M is RP2.

Proof. Let {pi} be the points of C where C or M are singular. Then ∂U is ob-tained from the disjoint union of the links L(pi, M) by attaching handles [0, 1]×S1.Attaching handles never creates an RP2.

Next we look at divisor–to–point contractions.

Proposition 5.5. Let M be a 3-dimensional PL-manifold and F ⊂ M a compact2-complex with only finitely many singular points. Let F ⊂ IntU ⊂ M be a regularneighborhood of F . Its topological normalization F (cf. 1.5) can be written as F (2)]F (1) where F (2) is a compact 2-manifold and F (1) is a 1-complex. Then

dim H1(F (2), Q) ≤ dim H1(∂U, Q),


52 JANOS KOLLAR

and strict inequality holds unless every connected component of F (2) is one of thefollowing:

(1) S2 or RP2,(2) a one-sided S1 × S1,(3) a one-sided Klein bottle whose neighborhood is not orientable.Moreover, if a connected component of ∂U is an RP2, then F has a subcomplex

which is a 2–sided RP2 in M .

Proof. Pick a point P ∈ F whose link in F consists of at least 2 circles. LocallyF looks like the cone over parallel plane sections (z = ai) ∩ (x2 + y2 + z2 = 1) ofthe unit sphere in R3 (plus a few 1-cells). By a homotopy we can replace this bythe parallel plane sections of the unit ball (z = ai) ∩ (x2 + y2 + z2 ≤ 1) and theinterval [mini{ai}, maxi{ai}] on the z-axis. This does not change the boundary ofthe regular neighborhood. Thus we may assume that F (2) → M is an embedding.

Let us take a point or a 1-cell e in F (1). If e does not intersect the rest of F , thena regular neighborhood of e is a 3-cell. e can be deleted from F without changingthe inequality.

If e intersects the rest of F in one endpoint only, then we can delete e from Fwithout changing the regular neighborhood.

If e intersects the rest of F at both endpoints, then removing e creates a new2-complex F ′, and F (2) = F ′(2). Let F ′ ⊂ IntU ′ be its regular neighborhood. ∂Uis obtained from ∂U ′ by attaching a handle [0, 1]× S1. Thus H1(∂U) ≥ H1(∂U ′),and it is sufficient to verify our inequality for F ′.

At the end we are reduced to the situation when F is the disjoint union ofembedded 2-manifolds, and it is sufficient to check the inequality for each connectedcomponent of F separately. ∂U → F is a 2-sheeted cover, thus H1(∂U) ≥ H1(F )with equality only if F ∼ S2, F ∼ RP2, F ∼ S1 × S1 ∼ ∂U or F and ∂U are bothKlein bottles.

RP2 cannot be obtained by attaching a handle to something else and a 1–sidedRP2 has S2 as the boundary of its regular neighborhood.

Proposition 5.6. Let M be a 3-dimensional PL-manifold and F ⊂ M a compact2-complex with only finitely many singular points. Let 0 ∈ N be obtained from M bycollapsing F to a point. Assume that N is a 3-manifold. Then M can be obtainedfrom N by repeated application of the following operations:

(1) taking connected sums of connected components,(2) taking the connected sum with S1 × S2,(3) taking the connected sum with S1×S2,(4) taking the connected sum with RP3.

Proof. We use the notation of (5.5) and of its proof. Let F ⊂ IntU ⊂ M and0 ∈ IntV ⊂ N be regular neighborhoods such that U = f−1(V ). Then ∂U = ∂V .Since N is a manifold, this implies that ∂U is a union of 2-spheres. We also seethat N is obtained from M \ IntU by attaching a 3-ball to each S2 in ∂U .

As in the proof of (5.5) we may assume that F (2) → M is an embedding.If e is a point or a 1-cell in F (1) which intersects the rest of F in zero or one

point only, then we can delete e from F .If e intersects the rest of F at both endpoints, then removing e creates a new

2-complex F ′ such that F = F ′. Let F ′ ⊂ IntU ′ be its regular neighborhood. ∂Uis obtained from ∂U ′ by attaching a handle [0, 1]× S1. The two ends {0}× S1 and



{1} × S1 cannot attach to the same connected component of ∂U ′ since that wouldcreate a torus or a Klein bottle in ∂U . Thus ∂U ′ has one more copy of S2 than ∂U .

N is obtained from N ′ by collapsing the image of e to a point, hence N and N ′

are homeomorphic by (5.2).At the end we are reduced to the situation when F is the disjoint union of

embedded copies of S2 and RP2. An S2 is necessarily 2-sided. Removing it fromF corresponds to taking connected sums of connected components (if S2 separatesM) or to taking the connected sum with S1×S2 or S1×S2 (if S2 does not separateM) (cf. [Hempel76, Chap. 3]).

If RP2 is 2-sided, then the boundary of its regular neighborhood consists of twocopies of RP2, so this cannot happen. A 1-sided RP2 corresponds to taking theconnected sum with RP3.

Corollary 5.7. Let f : X → Y be a morphism of real algebraic 3-folds. Assumethat

(1) X(R) and Y (R) are PL-manifolds, and(2) Ex(f) is a geometrically irreducible normal surface which is contracted to a

point.

Then X(R) can be obtained from Y (R) by repeated application of the followingoperations:

(3) removing an isolated point from Y (R),(4) taking connected sums of connected components,(5) taking the connected sum with S1 × S2,(6) taking the connected sum with S1×S2,(7) taking the connected sum with RP3.

Proof. If Ex(f)(R) = ∅, then the image of Ex(f) is an isolated real point of Y (R)which has to be thrown away to obtain X(R). If Ex(f)(R) 6= ∅, then isolated pointsof X(R) correspond to isolated points of Y (R), hence they can be ignored.

Let M be the topological normalization of X(R) \ (isolated points), N the topo-logical normalization of Y (R) \ (isolated points) and F the preimage of Ex(f)(R)in M . F is a 2-complex with isolated singularities since Ex(f) is normal.

Thus (5.7) follows from (5.6).

Complement 5.8. It is worthwhile to note that condition (5.7.2) can be weakenedto:

(2′) Ex(f) contains a unique geometrically irreducible surface S. S has onlyisolated singularities and S is contracted to a point by f .

It would be very useful to have a version of (5.7) which works if Ex(f) is anirreducible but nonnormal surface.

In the topological version (5.6) essentially nothing can be said if F is allowedto become an arbitrary compact 2-complex. For instance, let M be an arbitrarycompact 3-manifold and F the 2-skeleton of a triangulation of M . Then N is theunion of copies of S3 (one for each 3-simplex).

This example usually cannot arise as the real points of an algebraic surface, butit is not hard to modify it by approximating each simplex with a sphere to get thefollowing proposition. (This is not used in the sequel and so no proof is given here.)


54 JANOS KOLLAR

Proposition 5.9. Let M be a compact differentiable manifold of dimension n.Then there is a smooth real algebraic variety X and a morphism f : X → Ywith the following properties:

(1) X(R) ∼ M ,(2) Y (R) is a disjoint union of copies of Sn,(3) Ex(f) is a geometrically irreducible divisor and Ex(f)(R) is a union of copies

of Sn−1 intersecting transversally.

6. The gateway method

At the beginning of the MMP, divisorial contractions were considered to be theeasily understandable steps of the program and flips the hard one. Lately, however,more and more questions require a detailed understanding of all the steps of theMMP. A fairly complete description of all flips is known [Kollar-Mori92], but itseems very difficult to obtain a list of all divisorial contractions. One can try tostudy the MMP in two basic ways:

6.1 (Analysis of the MMP). Starting with a projective variety X , let us run theMMP. We obtain a sequence of birational maps

X = X0 99K X1 99K · · · 99K Xi 99K Xi+1 99K · · · 99K X∗.

Assume that X has some nice property that we would like to preserve. We needsome way of proving that X∗ also has this property, at least under some additionalassumptions. One way is to prove this directly, by analyzing each step of the MMP.This sometimes requires knowing each step of the MMP, and even in dimension 3the list is not yet available. Still there are many results that can be established thisway, for instance the existence of the MMP itself. In this approach one starts witha variety X and tries to understand every possible way an MMP can start with X .This is oftentimes manageable if X has only mild singularities.

Another way is to look at each step of the MMP backwards. In dimension 3 wehave a pretty good description of the possible singularities that arise in the course ofan MMP. Thus we can start with a variety Y and try to understand every possibleway an MMP can end with Y . This also seems rather hard. Even the case when Yis smooth is not at all understood, but in some other cases this approach has beencarried through (cf. [Kawamata96]). It seems that this method is easier to applywhen X is fairly singular.

The gateway method attempts to solve the original problem in an intermediateway. In the above chain of maps there is a smallest index i such that Xi is still“nice” but Xi+1 is not. Hence Xi 99K Xi+1 is a “gateway” through which theprocess leaves the set of “nice” varieties. Analyzing these “gateways” should beeasier since the direct approach tends to work for the nice variety Xi and thebackwards method tends to work for the more complicated singularities of Xi+1.

Once such a list of “gateways” is obtained, it is a matter of checking the list tosee if some additional properties ensure that these steps do not happen.

One of the simplest examples where these ideas yield a nontrivial result is thefollowing.

Example 6.2. Assume that we want to stay within the class of varieties of index1 (3.1). In this case there is only one gateway:



Proposition. Let f : X 99K X ′ be a step of the 3-dimensional MMP where X hasindex 1 but X ′ has higher index. Then f is the contraction of a divisor E ⊂ Xto a point. Furthermore, E ∼= P2, X is smooth along E and E has normal bundleOE(−2).

This result is a special case of [Mori88] and [Cutkosky88], though they did notapproach this from the point of view of gateways. A proof along the lines suggestedby the gateway method is not hard to construct, but this is not any shorter thanthe direct proofs.

As a consequence we obtain:

Corollary. Let X be a projective 3-fold with index 1 terminal singularities. Assumethat X does not contain any surface S ⊂ X which admits a birational morphismonto P2. Then each step of the MMP starting with X is a projective 3-fold withindex 1 terminal singularities.

Unfortunately the above condition needs to be checked for every surface S, evenfor very singular ones. Thus in practice this does not seem to be a useful observation.

6.3. Our aim is to develop a similar theory for real algebraic threefolds. Thus wehave to decide which varieties are “nice” and then describe all possible gatewaysthrough which the MMP can leave the class of “nice” varieties.

(4.7) naturally suggests a topological choice: X is “nice” if X(R) or X(R) isa 3-manifold, maybe with some additional properties. This was my first attempt,but I was unable to make it work. The main problem seems to be that, as thecomputations of [Kollar97b] show, there is basically no relationship between thealgebraic complexity of a terminal singularity 0 ∈ X and the topological complexityof its real points X(R).

Eventually I settled on a completely algebraic choice: X is nice if KX is Cartieralong X(R) (equivalently, if X has index 1 along X(R)). There are two mainreasons for adopting this definition:

(1) Most complications of 3-dimensional birational geometry come from the ap-pearance of points of index > 1. Hence this is likely to be the right choicealgebraically.

(2) One of the first things I realized was that under this condition there are noflips. Indeed, flips need higher index singular points to exist. If we have onlyindex 1 points along X(R), then all higher index points appear in conjugatepairs. A look at the list of flips [Kollar-Mori92] shows that the singularitiesappearing along a flipping curve are always asymmetrical.

Thus our task is to get a list of all steps f : Y → X of the MMP over R such thatKY is Cartier along Y (R). The case of divisor–to–curve contraction is relativelyeasy. Most of the work is devoted to studying the divisor–to–point contractions.Let 0 ∈ X(R) be the point in question. The existence of f is local in the Euclideantopology. I will go through the classification (up to real analytic equivalence) of3-dimensional terminal singularities over R and for each try to describe all possiblef : Y → X .

There is one subtle point here: the condition of Q-factoriality (3.1) is not pre-served under analytic equivalence. Thus first we need to develop a notion of “ex-tremal contraction without Q-factoriality”.


56 JANOS KOLLAR

Definition 6.4. Let X be a normal variety over a field K such that KX is Q-Cartier. A proper birational morphism f : Y → X is called an elementary extrac-tion of X if

(1) Y is normal and KY is Q-Cartier.(2) The exceptional set Ex(f) contains a unique K-irreducible divisor E.(3) −KY is f -ample.If we start with Y and construct f : Y → X , then f is usually called an elemen-

tary contraction of Y .We can write KY ≡ f∗KX + a(E, X)E where a(E, X) is the discrepancy of E.

Thus −a(E, X)E is f -ample. An exceptional divisor can never be relatively ample(cf. [Kollar-Mori98, 3.35]), thus a(E, X) > 0 and so −E is f -ample. This impliesthat Ex(f) = Supp E.

f(E) is also called the center of f on X .

A crucial property of elementary extractions is that they are determined by theirexceptional divisors:

Proposition 6.5. Let X be a normal variety over a field K such that KX is Q-Cartier. Let fi : Yi → X be elementary extractions with exceptional divisors Ei ⊂ Yi

for i = 1, 2. Assume that E1 and E2 correspond to each other under the birationalmap f−1

2 ◦ f1 : Y1 99K Y2. Then Y1 and Y2 are isomorphic (over X).

Proof. Let φ : f−12 ◦ f1 : Y1 99K Y2 be the composition. φ is birational, and Ex(φ),

Ex(φ−1) have codimension at least 2. Furthermore, KY1 and KY2 =φ∗(KY1) are rel-atively ample. Thus φ is an isomorphism by an argument of [Matsusaka-Mumford64,p.671].

In some sense this gives a way of enumerating all elementary extractions of X .We try to list all exceptional divisors over X and for each construct the correspond-ing unique elementary extraction. Usually there are infinitely many elementaryextractions for a given X and there does not seem to be an easy way to predict forwhich divisors the corresponding elementary extraction exists.

The next definition singles out a special class of elementary extractions, by re-stricting the singularities allowed on Y . The aim is to formalize a special case ofthe gateway method: we assume that Y is “nice”.

Definition 6.6. Let X be a normal variety over a field K such that KX is Q-Cartier. Assume for simplicity that X has only isolated singularities. A properbirational morphism f : Y → X is called a gateway–extraction or g–extraction if

(1) f is an elementary extraction with exceptional divisor E ⊂ Y .(2) Y has terminal singularities.(3) KY and E are Cartier at every K-point of Ex(f).

If Y has nonisolated singularities, then (3) should be replaced by(3′) KY and E are Cartier at the generic point of every geometrically irreducible

K-subvariety of Ex(f).If we start with Y and construct f : Y → X , then f is usually called a g–

contraction of Y .

The main technical aim of this article is to obtain a list of g–extractions forthreefolds with terminal singularities. The project turns out to be feasible since



the discrepancy a(E, X) of E (3.3) is always quite small. We have no a priori proofof this, but in every case the study of low discrepancy divisors leads to a description.

The relationship between low discrepancy divisors and g–extractions rests on thefollowing easy observation:

Proposition 6.7. Let X be a normal variety over a field K such that KX is Q-Cartier. Let f : Y → X be a g–extraction with exceptional divisor F ⊂ Y . Let Ebe a geometrically irreducible K-divisor over X such that centerX E ⊂ centerX F .Then

a(E, X) ≥ a(E, Y ) + a(F, X).

Proof. Let g : Z → X be a proper birational morphism such that centerZ E isa divisor on Z. We may assume that the induced rational map h : Z 99K Y is amorphism. h(E) is a geometrically irreducible K-subvariety of Y which is containedin Ex(f). Write

KZ ≡ g∗KX + a(E, X)E + (other exceptional divisors),KZ ≡ h∗KY + a(E, Y )E + (other exceptional divisors),KY ≡ f∗KX + a(F, X)F, andh∗F ≡ cE + (other exceptional divisors),

where c > 0 since h(E) ⊂ Ex(f) = Supp F and c is an integer by (6.6.3). Makingthe substitutions we obtain that a(E, X) = a(E, Y ) + c · a(F, X) ≥ a(E, Y ) +a(F, X).

The same method also proves the following result:

Proposition 6.8. Let X be a normal variety over a field K such that KX is Q-Cartier. Let f : Y → X be a morphism with exceptional divisor F =

⋃Fi ⊂

Y . Assume that Y has terminal singularities and KX and F are Cartier at thegeneric point of every geometrically irreducible K-subvariety of Ex(f). Assumethat mini{a(Fi, X)} ≥ 0.

Let E be a geometrically irreducible K-divisor over X such that centerX E ⊂⋃i centerX Fi. Then

a(E, X) ≥ a(E, Y ) + mini{a(Fi, X)}.

Corollary 6.9. Let X be a normal variety over a field K such that KX is Q-Cartier. Let f : Y → X be an elementary extraction with exceptional divisorE ⊂ Y . Assume that E is geometrically irreducible and a(E, X) ≤ 1.

Then either f : Y → X is a g–extraction, or X has no g–extractions whosecenter contains f(E).

Proof. Let g : Z → X be a g–extraction of X whose center contains f(E) and F ⊂ Zis the exceptional divisor. Then a(E, X) ≥ a(E, Z)+ a(F, X). If a(E, Z) = 0, thencenterZ E is a divisor which is contained in F . Since F is an irreducible divisor,centerZ E = F , hence Y = Z by (6.5). Otherwise a(E, Z) ≥ 1 which would forcea(F, X) ≤ 0. This contradicts (6.4).

Remark 6.10. This corollary gives a very efficient way of finding all g–extractionsof a given X in some cases. We have to find one geometrically irreducible divisorE such that a(E, X) ≤ 1 and construct the corresponding elementary extractionf : Y → X . Then it is usually easy to determine the singularities of Y .


58 JANOS KOLLAR

[Markushevich96] proved that if 0 ∈ X is a terminal threefold singularity which isnot smooth, then there is a divisor E over K with centerX E = {0} and a(E, X) ≤ 1.Thus there is always such an irreducible K-divisior, but it may not be geometricallyirreducible. Still, in many cases we are able to apply (6.9) directly.

In the remaining cases we show that there is always a geometrically irreducibledivisor E with centerX E = {0} and a(E, X) ≤ 3. This is still very useful, thanksto the following:

Corollary 6.11. Let X be a normal variety over a field K such that KX is Cartier.Let f : Y → X be an elementary extraction with exceptional divisor E ⊂ Y .Assume that E is geometrically irreducible. Let g : Z → X be any g–extractionwith exceptional divisor F whose center contains f(E).

Then either g = f or a(F, X) ≤ a(E, X)− 1.

Proof. By (6.7), a(E, X) ≥ a(E, Z) + a(F, X). If a(E, Z) = 0, then E and Fcorrespond to each other, hence Y = Z by (6.5). Otherwise a(E, Z) ≥ 1, thusa(F, X) ≤ a(E, X)− 1.

7. Small and divisor–to–curve contractions

In this section we look at those steps f : X → Y of the MMP over R which areeither small contractions or contract a divisor to a curve. The two cases can betreated together in the following setting:

Notation 7.1. Let K be a field of characteristic 0. Let X be a 3-fold over K withterminal singularities and f : X → Y a proper birational morphism over K suchthat −KX is f -ample and f∗OX = OY . Let 0 ∈ Y (K) be a closed point such thatdim f−1(0) = 1.

Under these assumptions R1f∗OX = R1f∗OX(KX) = 0 by the generalizedGrauert–Riemenschneider vanishing theorem (see, for instance, [CKM88, 8.8] or[Kollar-Mori98, 2.68]).

In keeping with the principles of the gateway method, we are interested in thecase when KX is Cartier at all points of X(K). The following theorem gives acomplete description of such contractions:

Theorem 7.2. Assume the notation and assumptions as in (7.1). Assume in ad-dition that KX is Cartier at all points of X(K). Then Y is smooth at 0 and onecan choose local (analytic or formal) coordinates (x, y, z) at 0 ∈ Y such that X isthe blow up of the curve (z = g(x, y) = 0) ⊂ Y for some g ∈ K[[x, y]].

In particular, f cannot be small.

This theorem has some very useful consequences for the MMP over R:

Corollary 7.3. Starting with a projective variety X over R, let

X = X0 99K X1 99K · · · 99K Xifi99K Xi+1

be the beginning of an MMP over R. Assume that KXj is Cartier at all points ofXj(R) for j ≤ i. Then the induced maps between the sets of real points

X(R) = X0(R) → · · · → Xi(R)fi→ Xi+1(R)

are everywhere defined.



Proof. The only steps of the MMP over R which are not everywhere defined arethe flips of small contractions (4.1). By (7.2) there are no flips in the sequence.

The topological behavior of divisor–to–curve contractions can also be determinedusing (7.2):

Theorem 7.4. Let X be a proper 3-fold over R with terminal singularities suchthat KX is Cartier at all points of X(R) and X(R) is a 3-manifold. Let f : X → Ybe a proper birational morphism over R such that −KX is f -ample and f∗OX = OY .Assume that dim f−1(y) ≤ 1 for every y ∈ Y . Then either

(1) f is R-small (that is, f : X(R) → Y (R) is an isomorphism outside codimen-sion 2 sets), or

(2) X(R) contains a 1-sided torus or Klein bottle with nonorientable neighbor-hood.

Proof. By (7.2), there is a real curve D ⊂ Y such that Y is smooth along Dand X = BDY (at least in a neighborhood of Y (R)). Pick 0 ∈ D(R) and let(z = g(x, y) = 0) be a local equation of D. By (7.5), either D is smooth at 0or X has a unique singular point over 0 with local equation st = g(x, y), which isequivalent to s2−t2−g(x, y) = 0. These are of type cA−

>1 or cA1 in the classificationof [Kollar97b]. If the origin is an isolated point of the real zero set (g = 0), thenX(R) \ f−1(0) → Y \ {0} is one–to–one near 0, hence f is R-small near 0.

Otherwise the real zero set (g = 0) is homeomorphic to a cone over an evennumber (say 2r) of points and from [Kollar97b, 4.4] we see that the real zero set(s2 − t2 − g(x, y) = 0) is homeomorphic to a cone over a surface of genus r − 1.Thus r = 1 and so D(R) is a PL–manifold near 0. Hence D(R) is the disjoint unionof some isolated points and some copies of S1.

If D(R) is finite, then f is R-small. Otherwise D(R) has a connected componentM ∼ S1. Let E ⊂ X be the exceptional divisor of f . By explicit computationwe see that E(R) → D(R) is an S1-bundle. Hence there is a unique connectedcomponent N ⊂ E(R) such that N is an S1-bundle over M . Thus N is either atorus or a Klein bottle. N is 1-sided with nonorientable neighborhood, since thesehold locally for the blow up of a smooth curve in a smooth 3-fold.

Example 7.5. Set Y = A3 with coordinates (x, y, z). Let X be the blow up of thecurve (z = g(x, y) = 0) ⊂ Y . Then X has a unique singular point which is givenby an equation st− g(x, y) = 0.

Corollary 7.6. Starting with a projective variety X over R, let

X = X0 99K X1 99K · · · 99K Xifi99K Xi+1

be the beginning of an MMP over R. Assume that

(1) KXj is Cartier at all points of Xj(R) for j ≤ i,(2) Xj(R) is a PL-manifold for j ≤ i,(3) X(R) satisfies the conditions (1.7).

Then:

(4) The induced maps between the sets of real points fj : Xj(R) → Xj+1(R) areeverywhere defined for j ≤ i,


60 JANOS KOLLAR

(5) For every j ≤ i + 1, there is a finite set Sj ⊂ Xj(R) such that Xj(R) \ Sj ishomeomorphic to an open subset of X(R),

(6) The smooth part of Xi+1(R) also satisfies the conditions (1.7).

Proof. The steps of an MMP are everywhere defined by (7.3).If g : U → V is any divisorial contraction over R, then Ex(g−1)(R) is finite unless

g is a divisor–to–curve contraction which is not R-small.Let fj be the first divisor–to–curve contraction in the sequence which is not R-

small. By the above remark, (5) holds for j. By (7.4), Xj(R) contains a surfaceF which is either a 1-sided torus or Klein bottle with nonorientable neighborhood.We can move F away from any finitely many points, thus by (5) X(R) also containsa 1-sided torus or Klein bottle with nonorientable neighborhood. This is a contra-diction. Hence among the steps there is no divisor–to–curve contraction which isnot R-small. This gives (5) and (6).

The proof of (7.2) relies on two results:

Proposition 7.7 ([Cutkosky88, Thm.4]). (7.2) holds if K is algebraically closedand C is irreducible.

Lemma 7.8 (cf. [Mori88, 1.14]). Let f : X → Y be a proper morphism and 0 ∈ Ya closed point such that dim f−1(0) = 1. Set red f−1(0) = C =

⋃Ci.

(1) If R1f∗OX = 0, then C is a tree of smooth rational curves.(2) Let D be a Q-Cartier Weil divisor on X such that D is Cartier at all but

finitely many points of f−1(0). Assume that (D · Ci) < 0 for every i andR1f∗OX(D) = 0. Then −1 ≤ (D · Ci) < 0 for every i and D is not Cartierat the singular points of C.

Proof. By replacing Y with a neighborhood of 0, we may assume that every fiberof f has dimension at most 1.

Let G be a sheaf on X such that R1f∗G = 0 and Q = G/F is a quotient of Gwhose support is in f−1(0). We get an exact sequence

R1f∗G → R1f∗Q → R2f∗F.

The left hand side is zero by assumption and the right hand side is zero since everyfiber of f has dimension at most 1. Thus R1f∗Q = 0.

Applying this with G = OX and Q = OC we conclude that H1(C,OC) =R1f∗OC = 0, hence C is a tree of smooth rational curves. This proves (1).

In order to see the second part, we may assume that the residue field of 0is algebraically closed. Then a point P ∈ C is singular iff there are at least 2irreducible components through P .OX(D)⊗OCi is a rank one locally free sheaf except possibly at the points where

D is not Cartier. Let Li denote its quotient by the torsion subsheaf. Then Li is aninvertible sheaf and we have a surjection OX(D) → Li. Applying R1f∗ we obtainas above that H1(Ci, Li) = 0. Thus deg Li ≥ −1.

On the other hand, for every m > 0 we have an injection

Lmi∼= (OX(D)⊗m ⊗OCi)/(torsion) ↪→ (OX(mD)⊗OCi)/(torsion).

If mD is Cartier, then the right hand side has negative degree, thus Lmi has negative

degree. Therefore deg Li = −1 for every i. Furthermore, m(D · Ci) ≥ m deg Li =−m, so (D · Ci) ≥ −1.



Set M := (OX(D) ⊗OC)/torsion). H1(C, M) = 0 as above. We have an exactsequence

0 → M →∑

Li → Q → 0,

where Q is supported at the singular points of C. Taking cohomologies, we concludethat H0(C, Q) = 0, thus Q = 0.

If D is Cartier at a singular point P of C, then M is locally free at P andM → ∑

Li cannot be surjective at P (it is not even surjective when tensored withthe residue field at P ).

Corollary 7.9. Assume the notation and assumptions as in (7.1). Then C is atree of smooth rational curves and KX is not Cartier at the singular points of C.

Proof. Apply (7.8) with D = KX . R1f∗OX = R1f∗OX(KX) = 0 by (7.1).

7.10 (Proof of (7.2)). The assumptions and conclusions are local near 0, thus wemay replace Y by a suitable analytic or formal neighborhood of 0.

By (7.9), C is a connected tree of smooth rational curves. Gal(K/K) acts onC, thus C either has a singular K-point or a geometrically irreducible componentdefined over K.

If P ∈ C is a singular point, then KX is not Cartier at P by (7.9), but ifP ∈ X(K), then KX is Cartier at P by assumption. Thus C cannot have asingular K-point.

Let C0 ⊂ C be a geometrically irreducible component defined over K. LetH ⊂ X be a divisor defined over K which intersects all irreducible components ofC \ C0 transversally but is disjoint from C0. (In order to do this, we may needto replace XK with a smaller analytic neighborhood of C.) By the basepoint freetheorem (cf. [CKM88, 9.3] or [Kollar-Mori98, 3.24]), a large multiple of H definesa morphism X → Y ′ → Y such that C0 is contracted to a point in Y ′. If (7.2)holds for X → Y ′, then KX is Cartier along C0. By (7.9) this implies that C0 is aconnected component of C. On the other hand, C is connected since f∗OX = OY .Thus C = C0 and Y ′ = Y .

Therefore it is sufficient to prove (7.2) under the additional assumption that Cis geometrically irreducible.

First we show that (7.2) holds if KX is Cartier along C. By (7.7), Y is smoothat 0 and X = BDY where D ⊂ Y is a curve of embedding dimension 2. D isthe image of the exceptional divisor of f , hence D is defined over K. Since D hasembedding dimension 2, its ideal is of the form (z, g(x, y)).

Finally we show that KX is Cartier along C. We start with the case whenK = R. Let P1, P1, . . . , Pk, Pk be all the conjugate pairs of points of index > 1(3.1). At each Pi pick a local member Di ∈ |KX | such that C ∩Di = Pi. (In orderto do this, we again may need to replace XK with a smaller analytic neighborhoodof C.) Let Di be the conjugates. Set D =

∑Di. Let m > 1 be the smallest natural

number such that mD is Cartier. D−D is a Weil divisor and OX(m(D−D)) ∼= OX

since the Picard group of a neighborhood of C is isomorphic to H2(C(C), Z) (cf.[Kollar-Mori98, 4.13]).

Corresponding to 1 ∈ H0(X,OX) we obtain an m-sheeted cyclic cover π : X →X (cf. [CKM88, 8.2.2] or [Kollar-Mori98, 2.52]) which is unramified outside thepoints of index > 1 (3.1). Thus KX = π∗KX and X has index 1 terminal singulari-ties. Let f : X → Y be the Stein factorization of X → Y . By the already discussed


62 JANOS KOLLAR

index 1 case, Y is smooth and one can choose local analytic coordinates (x, y, z) at0 ∈ Y such that X is the blow up of the curve (z = g(x, y) = 0) ⊂ Y .

The group Zm acts on f : X → Y and the quotient is f : X → Y . If mult0 g ≥ 2,then X has a unique singular point (7.5), which is necessarily fixed by the Zm-action.Thus X would have a unique point (of index m) which is the quotient of a singularpoint. On the other hand, the index > 1 singularities of X come in conjugatepairs. Therefore X is smooth and f : X → Y is the blow up of a smooth curve(z = y = 0) ⊂ Y .

We can choose local coordinates (x, y, z) on Y such that the action is

(x, y, z) 7→ (εax, εby, εcz)

where ε is a primitive mth root of unity and C = (y = z = 0). The correspondingaction on X has two fixed points (or a fixed curve) and the corresponding quotientsare

C3/ 1m (a, b− c, c) and C3/ 1

m (a, b, c− b).If terminal, these are both of type cA0/n on the list (3.15). A simple checkingshows that both of these cannot be simultaneously terminal.

If K is arbitrary, we can still proceed as above if we can find local divisorsDi ∈ |KX | at the index > 1 points such that (C ·∑ Di) = 0. Finding the Di needsa little case by case analysis, and sometimes it can be done only after first takingan auxiliary cover. It is probably easier to observe that there can be at most 2points of index > 1 along C (see, for instance, [CKM88, 14.5.5]), thus in fact theonly case we need to handle is when there is precisely one pair of conjugate pointsof index > 1.

8. Proof of the main theorems

The determination of all divisor–to–point g–extractions is rather technical andlengthy. In this section a summary of the list of all g–extractions is stated, andthen used to prove the main theorems stated in the introduction. The proof of (8.2)is given in sections 9–11.

Notation 8.1. Let g(x1, . . . , xm) be a polynomial or power series and M a monomialin the xi. M ∈ g means that M appears in g with nonzero coefficient.

Theorem 8.2. Let 0 ∈ X be a three-dimensional terminal singularity over R.Then X has a g–extraction whose center contains the origin iff (0 ∈ X) is realanalytically equivalent to one of the following 7 types.

The list below contains all g–extractions f : Y → X with geometrically irreducibleexceptional divisor E = red f−1(0) and it indicates if there are other g–extractions.

(1) (cA0, smooth point) X = A3. There are precisely 2 types of g–extractions:(a) (point blow up) Y = B0A3 → A3 and E ∼= P2.(b) (curve blow up) Y = BCA3 → A3 where C ⊂ A3 is an irreducible, real

and locally planar curve. The exceptional divisor is a P1-bundle over C.(2) (cA0/2) X = A3/Z2, where the Z2-action on A3 is (x, y, z) 7→ (−x,−y,−z).

There is a unique g–extraction Y = B0A3/Z2 → A3/Z2 with exceptionaldivisor E ∼= P2.

(3) (cA+>0, mult0 g even) X = (x2 + y2 + g≥2m(z, t) = 0) where g2m(z, t) 6= 0

and m ≥ 1. There is a unique g–extraction Y = B(m,m,1,1)X → X withexceptional divisor E = (x2 + y2 + g2m(z, t) = 0) ⊂ P3(m, m, 1, 1).



(4) (cA+>0, mult0 g odd) X = (x2 +y2 +g≥2m+1(z, t) = 0) where g2m+1(z, t) 6= 0

and m ≥ 1. There is a g–extraction with geometrically irreducible ex-ceptional divisor iff there is a linear change of the (z, t)-coordinates suchthat g≥2m+1(z, t) =

∑2i+j≥4m+2 γijz

itj and γ2m+1,0 6= 0. The uniquesuch g–extraction is Y = B(2m+1,2m+1,2,1)X → X with exceptional divi-sor E = (x2 + y2 +

∑2i+j=4m+2 γijz

itj = 0) ⊂ P3(2m + 1, 2m + 1, 2, 1).G–extractions with geometrically reducible exceptional divisor have not beenfully enumerated; see (10.10).

(5) (cA+>0/2, mult0 g even) X = (x2 + y2 + g≥2m(z, t) = 0)/Z2, where g2m 6= 0,

m ≥ 1, and the Z2-action is (x, y, z, t) 7→ (−x,−y,−z, t). There is a g–extraction iff m is even and z2m, t2m ∈ g. The unique g–extraction is Y =B(m,m,1,1)X/Z2 and E = E/Z2, where X = (x2 + y2 + g≥2m(z, t) = 0) andE = (x2 + y2 + g2m(z, t) = 0) ⊂ P3(m, m, 1, 1).

(6) (cA+>0/2, mult0 g odd) X = (x2+y2+g≥2m+1(z, t) = 0)/Z2 where g2m+1(z, t)

6= 0, m ≥ 1, and the Z2-action is (x, y, z, t) 7→ (−x,−y,−z, t). Thereis a g–extraction iff g≥2m+1(z, t) =


itj and γ2m+1,0 6= 06= γ0,4m+2. The unique g–extraction is Y = B(2m+1,2m+1,2,1)X/Z2 andE = E/Z2, where X = (x2 + y2 +


itj) = 0) and E =(x2 + y2 +

∑2i+j=4m+2 γijz

itj = 0) ⊂ P3(2m + 1, 2m + 1, 2, 1).(7) (cE6) X = (x2 + y3 + (z2 + t2)2 + yg≥4(z, t)+ h≥6(z, t) = 0). There is no g–

extraction with geometrically irreducible exceptional divisor. G–extractionswith geometrically reducible exceptional divisor have not been fully enumer-ated; see (11.6).

Corollary 8.3. Let 0 ∈ X be a three-dimensional terminal singularity over R andf : Y → X a g–extraction with exceptional divisor E = red f−1(0). If E is geomet-rically irreducible, then E is normal.

Proof. Equations for E are given in (8.2). E ∼= P2 in the first two cases. In theremaining cases E is (or is the quotient of) a surface of the form

F := (x2 + y2 + p(z, t) = 0) ⊂ P3(r, r, s, 1).

All the singularities of F are contained in the (x = y = 0) line. Thus we get onlyfinitely many singularities if p is not identically zero, which is always the case in(8.2).

Theorem 8.4. Let Y be a projective real algebraic 3–fold with terminal singu-larities such that KY is Cartier along Y (R). Let n : Y (R) → Y (R) denote thetopological normalization and assume that Y (R) is a 3–manifold.

Let f : Y → X be a g–contraction (6.6) with exceptional divisor E. (ThusX is also a projective real algebraic 3–fold with terminal singularities.) Assumethat n−1(E(R)) does not have any irreducible component which is a 2-sided RP2, a1-sided torus or a 1-sided Klein bottle with nonorientable neighborhood in Y (R).

Then KX is Cartier along X(R) and X(R) is a 3–manifold. If E(R) 6= ∅, then(f(E))(R) is finite and for every 0 ∈ (f(E))(R) the germ (0 ∈ X) is real analyticallyequivalent to one of the following 4 types.

(1) (cA0) X is smooth at 0. There are 2 types of g–extractions:(a) (point blow up) Y = B0X and E ∼= P2.


64 JANOS KOLLAR

(b) (curve blow up) Y = BCX where C ⊂ X is an irreducible, real andlocally planar curve with only finitely many real points.

(2) (cA+>0, mult0 g even) X = (x2 + y2 + g≥2m(z, t) = 0) where g2m(z, t) 6= 0

and g≥2m is not everywhere negative in a punctured neighborhood of 0. Y =B(m,m,1,1)X and E = (x2 + y2 + g2m(z, t) = 0) ⊂ P3(m, m, 1, 1).

(3) (cA+>0, mult0 g odd) X = (x2+y2+g≥2m+1(z, t) = 0) where g2m+1(z, t) 6= 0.

f : Y → X is either R-small (I do not have a complete list of these) orthere is a linear change of the (z, t)-coordinates such that g≥2m+1(z, t) =∑

2i+j≥4m+2 γijzitj and γ2m+1,0 is nonzero. In this coordinate system

Y = B(2m+1,2m+1,2,1)X and E = (x2 + y2 +∑

2i+j=4m+2 γijzitj = 0) ⊂

P3(2m + 1, 2m + 1, 2, 1).(4) (cE6) X = (x2 + y3 + (z2 + t2)2 + yg≥4(z, t)+h≥6(z, t) = 0) and f : Y → X

is R-small (I do not have a complete list of these).

Proof. First of all we know that f : Y → X appears on the list of (8.2). Weexclude several of the cases using the assumption that n−1(E(R)) does not haveany irreducible component which is a 2-sided RP2, a 1-sided torus or a 1-sided Kleinbottle with nonorientable neighborhood in Y (R).

The case when C(R) is 1–dimensional in (8.2.1.b) is excluded by (7.4). In case(8.2.2), E(R) is a 2-sided RP2 in Y (R).

If X has a singularity of type (8.2.3), then by [Kollar97b, 4.4] X satisfies theconclusions except when g≥2m is everywhere negative in a punctured neighborhoodof 0. In this case X(R) is a cone over a torus near 0. This gives only a 2-sidedtorus in X(R) and in Y (R) which is allowed. We proceed to prove, however, thatwe still get a 1-sided torus in Y (R) coming from the exceptional divisor E. Thiscontradicts our assumptions.

By (8.2.3) f : Y → X is the (m, m, 1, 1)-blow up. We distinguish two cases:General case: g2m(z, t) is negative on R2 \ {0}. The exceptional divisor E of the

above g–extraction is the weighted hypersurface

E = (x2 + y2 + g2m(z, t) = 0) ⊂ P(m, m, 1, 1).

Its canonical divisor is KE = OE(−2), thus E is orientable. The projection(x : y : z : t) 7→ (z : t) exhibits E as an S1-bundle over RP1, thus E ∼ S1 × S1.L(0 ∈ X(R)) is connected, thus E(R) ⊂ Y (R) is a 1-sided torus, a contradiction.

Special case: g2m(z, t) is not negative on R2\{0}. g2m(z, t) is the leading term ofg≥2m(z, t), which is negative on R2 \{0}. Thus g2m(z, t) is nonpositive on R2 \{0}.

We use the notation of (9.1) below. The t-chart on B(m,m,1,1)X is x21 + y2

1 +t−2m1 g(z1t1, t1). Set g′(z1, t1) := t−2m

1 g(z1t1, t1). Then g′(z1, t1) is strictly negativeoutside the z1-axis, and is not identically zero on the z1-axis. Thus g′(z1, t1) iseverywhere nonpositive with only finitely many zeros. At each zero of g′(z1, t1), Yhas a singular point of the same type we started with. By [Kollar97b, 4.4], Y (R) islocally a cone over a torus at these points, in contradiction to our assumption thatY (R) be a manifold. This completes the case (8.2.3).

If (8.2.4) holds, then X(R) is a 3–manifold by [Kollar97b, 4.4] and similarly for(8.2.7) using [Kollar97b, 4.9].

We still have to exclude the cases (8.2.5–6). E(R) is a 2–complex with finitelymany singular points. If one of the connected components of the link L(0 ∈ X(R)) isan RP2, then n−1(E(R)) contains a 2–sided RP2 by (5.5). The links L(0 ∈ X(R))were described in [Kollar97b, 5.9], and we obtain that they always have an RP2



component except possibly when g(z, t) is negative in a punctured neighborhoodof the origin. Then g has even multiplicity, so we are in case (8.2.5). We againdistinguish two cases:

General case: g2m(z, t) is negative on R2 \ {0}. The exceptional divisor E is theZ2 quotient of the weighted hypersurface

E = (x2 + y2 + g2m(z, t) = 0) ⊂ P(m, m, 1, 1).

We already determined that E(R) is a torus and a choice of orientation is given by(dy ∧ dz)/x. The Z2-action sends this to

(d(−y) ∧ d(−z))/(−x) = −(dy ∧ dz)/x,

hence E(R)/Z2 is not orientable. We conclude that one of the connected com-ponents of E(R) is a Klein bottle. (There may be other connected components.)The Klein bottle is 1-sided since E(R) is 1-sided. The regular neighborhood isnonorientable since its boundary, the link of 0 ∈ X(R), is again a Klein bottle.

Special case: g2m(z, t) is not strictly negative on R2 \ {0}.The same computation as in the (8.2.3) case above shows that this leads to a

nonmanifold point on Y (R), which contradicts the assumptions.

8.5 (Proof of (1.10) and (1.13)). Starting with X , let us run the MMP over R.We get a sequence

X = X0 99K X1 99K · · · 99K Xifi99K Xi+1.

Assume by induction that (1.10) holds for Xj for j ≤ i and (1.13) holds for fj :Xj 99K Xj+1 for j ≤ i− 1.

We need to show that (1.10) holds for Xi+1 and (1.13) holds for fi : Xi 99K Xi+1.By (7.6) fj : Xj(R) → Xj+1(R) are everywhere defined for j ≤ i − 1 and

Xi(R) does not contain an RP2, a 1-sided torus or Klein bottle with nonorientableneighborhood. Furthermore, by (7.4), fi is either R-small or a divisor–to–pointcontraction.

By induction Xi has index 1 along Xi(R) and Xi(R) is a manifold. Thus fi isone of the cases listed in (8.4). These are exactly the ones allowed in (1.10) and(1.13).

8.6 (Proof of (1.2) and (1.8)). We follow the steps of an MMP over R, using(1.13). fi : Xi(R) → Xi+1(R) is a homeomorphism in cases (1.13.1–2) while (1.13.3)gives a connected sum with RP3.

In the cases (1.13.4) the exceptional divisor is normal by (8.3), hence we getvarious cases of (1.2) by (5.7).

Example 8.7. Consider the singularity X := (x2 + y2 + z2m+1 + t4m+2 = 0). The(2m+1, 2m+1, 2, 1)-blow up X1 → X is a g–extraction which is smooth along theR-points.

The (m, m, 1, 1)-blow up is another g–extraction whith one singular point(x2

1 + y21 + z2m+1

1 t1 + t2m+21 = 0) on the t-chart. After the (m + 1, m + 1, 1, 1)-

blow up we obtain a variety X2 → X which is smooth along its R-points.These two resolutions are indeed quite different. Using the methods of section

5, we see that X1(R) ∼ X(R) # RP3 and X2(R) ∼ X(R) # S1 × S2.


66 JANOS KOLLAR

9. cAx and cD-type points

In this section we begin to classify g–extractions (6.6) of terminal singularitiesover any field K. We mostly care about K = R, and so we do not fully discusssome examples which do not occur in the real case. The classification of 3–foldterminal singularities over nonclosed fields is done in [Kollar97b]. The results aresummarized in (3.15). We work through the list of the singularities. In most casesit is easy to see that there are no g–extractions. This is done by exhibiting anelementary extraction (6.4) which is not a g–extraction. If the discrepancy of theexceptional divisor is ≤ 1, then there are no g–extractions by (6.9).

In this section we deal with the cases cAx/2, cAx/4, cD, cD/2, cD/3. Amongterminal singularities these are somewhat esoteric but the proofs work well forthem: in each case (6.9) applies.

The remaining terminal singularities are considered in the next 2 sections. Insome cases much more complicated arguments are needed to classify all g–extrac-tions.

Definition 9.1 (Weighted blow-ups). Let x1, . . . , xn be coordinates on An. Theusual blow up of the origin is patched together from affine charts with morphismsof the form

xj = x′jx′i if j 6= i and xi = x′i.

We refer to this as the xi-chart .Let a1, . . . , an be a sequence of positive integers. For every 1 ≤ i ≤ n we can

define a morphism Πi : An → An by

xj = x′j(x′i)

aj if j 6= i and xi = (x′i)ai .

This morphism is birational iff ai = 1 and has degree ai in general. One can easilynotice that Πi is invariant under the Zai -action

(x′1, . . . , x′n) 7→ (ε−a1x′1, . . . , ε

−ai−1x′i−1, εx′i, ε

−ai+1x′i+1, . . . , ε−anx′n)

(where ε is a primitive aith root of unity) and it descends to a birational morphismπi,

Πi : An(x′1, . . . , x′n) → An(x′1, . . . , x

′n)/Zai

πi−→ An(x1, . . . , xn).Furthermore, these charts patch together to give a projective morphism

π : B(a1,...,an)An → An.

This is called the weighted blow up of An with weights a1, . . . , an.

Notation 9.2. In the proofs in sections 9–11 we use the following conventions.First we state the name of the singularity X from (3.15) and possibly some

other restrictions. Then we write down the normal form of the equation X =F (x, y, z, t)/ 1

r (bx, by, bz, bt) (3.13). Any restrictions on F are explained in detailhere.

Then we specify the weights (ax, ay, az, at) for a weighted blow up and write downthe equation of the birational transform of X on one of the charts on the weightedblow up. Before taking quotients, this has the form t−m

1 F (x1tax1 , y1t

ay

1 , z1taz1 , tat

1 ) ifwe use the t-chart. This is denoted by BX.

We need to take the quotient by 2 actions. First is the 1at

(−ax,−ay,−az, 1)-action coming from the weighted blow up. Second, the 1

r (bx, by, bz, bt)-action needsto be lifted to the (x1, y1, z1, t1)-space. In some cases this lifts as a Zr-action but in



other cases the actions combine into a Z(rat)-action. The quotient of BX by these2 actions is a chart on the weighted blow up of X ; it is denoted by BX .

All these can be done in 4 different charts. We chose the chart where the singu-larities are most visible or the discrepancy computation is the clearest.

Finally we compute the exceptional divisor of the blow up, the singularities ofBX and the discrepancy of the exceptional divisor.

9.3 (cAx/2). Normal form: ax2 + by2 + g≥4(z, t)/ 12 (1, 0, 1, 1), where ab 6= 0.

Weights for blow-up: (1,1,1,1).t-chart: ax2

1 + by21 + t−2

1 g≥4(z1t1, t1)/ 12 (0, 1, 0, 1).

Exceptional divisor: (t1 = ax21 + by2

1 = 0). Over K this is reducible and the twoirreducible components are (t1 =

√ax1 ±

√−by1 = 0). The Z2-action interchangesthese two, so on the quotient we get a geometrically irreducible exceptional divisor.

Singularity: The Z2-action has a fixed curve on BX: the intersection with the(x1 = z1 = 0)-plane. Thus we get a curve of nonterminal singularities on BX .

Discrepancy: π∗ dy∧dz∧dtx = t1

dy1∧dz1∧dt1x1

, so a(E, X) = 1.

9.4 (cAx/4). Normal form: ax2 + by2 + g≥2(z, t)/ 14 (1, 3, 1, 2), where ab 6= 0 and

g2(0, 1) = 0 for weight reasons.Weights for blow-up: (1,1,1,1).t-chart: ax2

1 + by21 + t−2

1 g≥2(z1t1, t1)/ 14 (3, 1, 3, 2).

Exceptional divisor: E := (t1 = ax21 + by2

1 + g2(1, 0)z21 = 0). E is geometrically

irreducible if g2(1, 0) 6= 0. If g2(1, 0) = 0, then E is reducible over K, and the twoirreducible components are (t1 =

√ax1 ±

√−by1 = 0). The Z4-action interchangesthese two, so on the quotient we get a geometrically irreducible exceptional divisorE.

Singularity: The origin is on BX since g2(0, 1) = 0 and it is a fixed point. Weget an index 4 point on BX .


dy1∧dz1∧dt1x1

, so a(E, X) = 1.

9.5 (cD4 main series). Normal form: x2 + f≥3(y, z, t), where we assume thatf3(y, z, t) is irreducible over K.

Weights for blow-up: (2,1,1,1).x-chart: x1 + x−3

1 f≥3(y1x1, z1x1, t1x1)/ 12 (1, 1, 1, 1).

Exceptional divisor: E := (x1 = f3(y1, z1, t1) = 0). E is geometrically irre-ducible by our assumption.

Singularity: The origin is a fixed point on BX , hence we get an index 2 pointon BX .

Discrepancy: π∗ dy∧dz∧dtx = x1 · dy1 ∧ dz1 ∧ dt1, so a(E, X) = 1.

9.6 (cD4/2 main series). Normal form: x2 + f≥3(y, z, t)/ 12 (1, 1, 0, 1), where we as-

sume that f3(y, z, t) is irreducible over K. However, for weight reasons z|f3(y, z, t),so this cannot happen.

9.7 (cD/3). Normal form: x2+f≥3(y, z, t)/ 13 (0, 1, 1, 2), where f3(0, 0, t) 6= 0. Since

this is not a cE point and for weight reasons, also f3(y, z, 0) 6= 0. We can writef3 = t3 + f3(y, z, 0).

Weights for blow-up: (2,1,1,1).x-chart: x1 +x−3

1 f≥3(y1x1, z1x1, t1x1)/ 12 (1, 1, 1, 1), and then take the Z3-action.

Lifting of the Z3-action: It lifts to 16 (3, 5, 5, 1).


68 JANOS KOLLAR

Exceptional divisor: E := (x1 = f3(y1, z1, t1) = 0). E is geometrically irre-ducible if f3(y, z, 0) is not a cube over K. If f3(y, z, 0) = −L(y, z)3 over K, thenE has three geometrically irreducible components (x1 = t1− ηL(y1, z1) = 0) whereη3 = 1. The Z6-action permutes these, so on BX we get a geometrically irreducibleexceptional divisor.

Singularity: The origin is a Z6-fixed point which has multiplicity 1 on BX . BXhas a terminal quotient singularity of index 6.

Discrepancy: π∗ dy∧dz∧dtx = x1 · dy1 ∧ dz1 ∧ dt1, so a(E, X) = 1.

9.8 (cD>4 and special cD4). Normal form: x2 + Q2(y, z, t)z + g≥4(y, z, t), whereQ2(y, 0, t) 6= 0. In the cD>4 we always have this form (with Q2(y, z, t) = y2). Inthe cD4-case we can achieve this form iff f3(y, z, t) has a simple linear factor overK.

Weights for blow-up: (2,1,2,1).z-chart: x2

1 + Q2(y1, z1, t1) + z−41 g≥4(y1z1, z

21 , t1z1)/ 1

2 (0, 1, 1, 1).Exceptional divisor: E := (z1 = x2

1 + Q2(y1, 0, t1) + g4(y1, 0, t1) = 0). E isgeometrically irreducible iff Q2(y1, 0, t1) is not a square over K or g4(y1, 0, t1) 6= 0.If Q2(y1, 0, t1) = −L1(y1, t1)2 (over K) and g4(y1, 0, t1) = 0, then E is reducibleover K, and the two irreducible components are (z1 = x1 ± L1(y1, t1) = 0). TheZ2-action interchanges these two, so E ⊂ BX is geometrically irreducible.

Singularity: The origin is a fixed point on BX , hence we get an index 2 pointon BX .

Discrepancy: π∗ dy∧dz∧dtx = 2z1

dy1∧dz1∧dt1x1

, so a(E, X) = 1.

9.9 (cD>4/2 and special cD4/2). Normal form: x2 + Q2(y, z, t)z + g≥4(y, z, t)/12 (1, 1, 0, 1).

Weights for blow-up: (2,1,2,1).z-chart: x2

1 +Q2(y1, z1, t1)+z−41 g≥4(y1z1, z

21 , t1z1)/ 1

2 (0, 1, 1, 1) and then take theZ2-action.

Lifting of the Z2-action: We get a pair of commuting Z2-actions on BX , givenby 1

2 (0, 1, 1, 1) and 12 (1, 0, 1, 0).

Singularity: The second action has a fixed curve on BX, so BX is singular alonga curve.

Exceptional divisor and discrepancy: As in the cD>4-case.

9.10 (cD-cases, conclusion). We have settled all the cD>4, cD/2 and cD/3 cases,they have no g–extractions by (6.9).

In the cD4 cases there are no g–extractions if f3 is irreducible or if it has a simplelinear factor over K. The only remaining case is when f3 is the product of 3 linearfactors which are conjugate over K.

This cannot happen when K = R (or more generally if K is real closed), so oversuch fields points of type cD, cD/2 and cD/3 do not have g–extractions.

The situation is more complicated over fields which do have cubic extensions, asthe following example shows. We have not classified all cases.

Example 9.11. Consider x2 + y3 + az3 + t6, where a ∈ K is not a cube. Theexceptional divisor of the (3, 2, 2, 1)-blow up is irreducible and has discrepancy 1.It has three points of index 2 which are conjugate over K, and no other singularities.Hence this is a g–extraction.



Example 9.12. We obtain an interesting example from the equation x2 +(y2 + z2)z + t5. The (2, 1, 2, 1)-blow up has terminal singularities (one with in-dex 2). The exceptional divisor E is singular along a curve.

10. cA-type points

In this section we study g–extractions of cA type terminal singularities. Theconventions of (8.1), (9.1) and of (9.2) are used throughout.

10.1 (cA0). (That is, smooth points.) Normal form: A3.The blow up of the origin is smooth with exceptional divisor E ∼= P2. a(E, X) =

2, and by (6.7) a(F, X) ≥ 2 for every exceptional divisor F with centerX F = {0}.Therefore by (6.11), the blow up of the origin is the only g–extraction.

The exceptional divisor is E ∼= P2 with normal bundle OP2(−1).

10.2 (cA0/n, n ≥ 2). Normal form: A3/ 1n (r,−r, 1), where (r, n) = 1 and 1 ≤ r ≤

n− 1.Weights for blow-up: (r, n− r, 1).x-chart: A3(x1, y1, z1)/ 1

r (1,−n,−1).Exceptional divisor: E := (x1 = 0). Geometrically irreducible and invariant

under the Zr-action.Lifting of the Zn-action: The Zn-action lifts to 1

n (1, 0, 0). Its invariants arex2 := xn

1 and y1, z1. The Zr-action descends to the quotient of the Zn-action asA3(x2, y1, z1)/ 1

r (n,−n,−1).Singularity: We obtain an index r point on the x-chart, and similarly an index

n− r point on the y-chart.Discrepancy: π∗dx ∧ dy ∧ dz = rxn

1 dx1 ∧ dy1 ∧ dz1 = rnx1dx2 ∧ dy1 ∧ dz1. Since

x1 = x1/n2 , we obtain that a(E, X) = 1/n.

Conclusion: The above blow up is the only possible g–extraction. If n ≥ 3, theneither r ≥ 2 or n− r ≥ 2, and we obtain a singular point of index ≥ 2 on BX .

If r = 2, then BX is smooth, the exceptional divisor is E ∼= P2 with normalbundle OP2(−2). BX → X is the unique g–extraction.

10.3 (cA1). Normal form: ax2 + by2 + cz2 + dtm, where abcd 6= 0.Weights for blow-up: (1,1,1,1).t-chart: ax2

1 + by21 + cz2

1 + dtm−21 .

Exceptional divisor: E := (t1 = ax21 + by2

1 + cz21 = 0) for m ≥ 3 and (t1 =

ax21 + by2

1 + cz21 + d = 0) for m = 2. E is geometrically irreducible.

Singularity: BX has exactly one singular point for m ≥ 4, it lies on the t-chart.BX is smooth for m = 2, 3.


dy1∧dz1∧dt1x1

, so a(E, X) = 1.Conclusion: The only g–extraction is this blow up. The singularities can be

resolved by repeatedly blowing up the unique singular point.

10.4 (cA1/2). Normal form: ax2 + by2 + czn + dtm/ 12 (1, 1, 1, 0), where abcd 6= 0

and min{n, m} = 2.Weights for blow-up: (1,1,1,1).z-chart: ax2

1 + by21 + czn−2

1 + dtm1 zm−21 .

Exceptional divisor: E := (z1 = ax21 + by2

1 + c = 0) for m ≥ 3, (z1 = ax21 +

by21 + dt21 = 0) for n ≥ 3 and (z1 = ax2

1 + by21 + c + dt21 = 0) for n = m = 2. E is

geometrically irreducible.


70 JANOS KOLLAR

Singularity: The 12 (1, 1, 1, 0)-action lifts to a 1

2 (0, 0, 1, 1)-action. Thus we get afixed curve where the blow up intersects the plane (z1 = t1 = 0).

Discrepancy: π∗ dy∧dz∧dtx = z1

dy1∧dz1∧dt1x1

, so a(E, X) = 1.Conclusion: The only possible g–extraction is this blow up. It has nonterminal

singularities, so this does not occur.

10.5 (cA1/n, n ≥ 3). Normal form: xy+czpn +dt2/ 1n (r,−r, 1, 0), where (r, n) = 1

and cd 6= 0.Weights for blow-up: (1,1,1,1).z-chart: x1y1 + czpn−2

1 + dt21/1n (r − 1, 1− r, 1,−1).

Exceptional divisor: E := (z1 = x1y1 + dt21 = 0). It is geometrically irreducible.Singularity: The Zn-action has an isolated fixed point at the origin on BX. Thus

BX has an index n point.Discrepancy: π∗ dy∧dz∧dt

x = z1dy1∧dz1∧dt1

x1, so a(E, X) = 1.

Conclusion: The only possible g–extraction is this blow up. It has a higher indexpoint, so this does not occur.

10.6 (cA−>1). Normal form: xy + g≥3(z, t).

Weights for blow-up: (1,1,1,1).t-chart: x1y1 + t−2

1 g≥3(z1t1, t1).Exceptional divisor: E := (t1 = x1y1 = 0). It has two geometrically irreducible

components.Singularity: Not important.Discrepancy: π∗ dy∧dz∧dt

x = t1dy1∧dz1∧dt1

x1, so a(E, X) = 1.

Conclusion: There are at least 2 geometrically irreducible divisors with discrep-ancy ≤ 1, so no g–extractions.

10.7 (cA>1/n, n ≥ 3 and cA−>1/2). Normal form: xy + g≥3(z, t)/ 1

n (r,−r, 1, 0),where (r, n) = 1.

Weights for blow-up: (1,1,1,1).t-chart: x1y1 + t−2

1 g≥3(z1t1, t1)/ 1n (r,−r, 1, 0).

Exceptional divisor: E := (t1 = x1y1 = 0). It is reducible and both irreduciblecomponents are geometrically irreducible and invariant under the Zn-action.

Singularity: Not important.Discrepancy: π∗ dy∧dz∧dt

x = z1dy1∧dz1∧dt1

x1, so a(E, X) = 1.

Conclusion: There are at least 2 geometrically irreducible divisors with discrep-ancy ≤ 1, so no g–extractions.

10.8 (cA+>1, mult0 g even). Normal form: ax2 + by2 + g≥2m(z, t), where m ≥ 2,

−ab is not a square and g2m 6= 0.Weights for blow-up: (m, m, 1, 1).t-chart: ax2

1 + by21 + t−2m

1 g≥2m(z1t1, t1).Exceptional divisor: E := (t1 = ax2

1 + by21 + g2m(z1, 1) = 0). It is geometrically

irreducible.Singularity: The t-chart on BX is singular only at points P corresponding to

the multiple roots of g2m(z, 1). The singularity at P again has type cA+>1, but the

multiplicity of the corresponding gP (z1, t1) is not necessarily even. The z chart issimilar.


dy1∧dz1∧dt1x1

, so a(E, X) = 1.x-chart: a + by2

1 + x−2m1 g≥2m(z1x1, t1x1)/ 1

m (1, 0,−1,−1).



Singularity: The fixed points of the Zm-action are along the y1-axis; this inter-sects BX in two points (0,

√−a/b, 0, 0) which are conjugate over K. Thus BXhas 2 index m terminal singularities which are conjugate over K. No other newsingular points. The y-chart is similar.

Conclusion: The only g–extraction is the above weighted blow up. The excep-tional divisor is geometrically irreducible with a pair of conjugate index m-points.The other singular K-points of BX are again of type cA+

>1 or cA1.

10.9 (cA+>1/2, mult0 g even). Normal form: ax2 + by2 + g≥2m(z, t)/ 1

2 (1, 1, 1, 0),where −ab is not a square and g2m 6= 0.

Weights for blow-up: (m, m, 1, 1).z-chart: ax2

1 + by21 + z−2m

1 g≥2m(z1, t1z1)/ 12 (1−m, 1−m, 1, 1).


1 + g2m(1, t1) = 0) is geometricallyirreducible.

Singularities: If m is odd, then (z1 = t1 = 0) intersects BX in a fixed curve ofthe Z2-action, thus we get a singular curve on BX .

If m is even, then on the z-chart the only Z2-fixed point is the origin. This isnot on BX iff z2m ∈ g.

t-chart: ax21 + by2

1 + t−2m1 g≥2m(z1t1, t1)/ 1

2 (1, 1, 1, 0).Singularities: On the t-chart the fixed point set is the t1-axis. This intersects

the exceptional divisor at the origin. This is not on the blow up iff t2m ∈ g.Discrepancy: π∗ dy∧dz∧dt

x = t1dy1∧dz1∧dt1

x1, so a(E, X) = 1.

x-chart: a + by21 + x−2m

1 g≥2m(z1x1, t1x1)/ 1m (1, 0,−1,−1) and we also need to

take the quotient by the Z2-action.Lifting the Z2-action: a + by2

1 + x−2m1 g≥2m(z1x1, t1x1)/ 1

2m (1, 0, m− 1,−1).Singularities: On the x-chart the fixed point set is the y1-axis. This intersects

BX at two points (0,±√−a/b, 0, 0). We get a conjugate pair of terminal singular-ities of index 2m on BX .

y-chart: Similar to the x-chart.Conclusion: ax2 + by2 + g≥2m(z, t)/ 1

2 (1, 1, 1, 0), where −ab is not a square, has ag–extraction iff m is even and z2m, t2m ∈ g2m(z, t). Under these assumptions, theunique g–extraction is the (m, m, 1, 1)-blow up.

10.10 (cA+>1, mult0 g odd). Normal form: ax2 + by2 + g≥2m+1(z, t), where m ≥ 1,

−ab is not a square and g2m+1 6= 0.Weights for blow-up: (s, s, 1, 1) for 1 ≤ s ≤ m, giving BsX → X .t-chart: ax2

1 + by21 + t−2s

1 g≥2m+1(z1t1, t1).Exceptional divisor: E := (t1 = ax2

1 + by21 = 0). It is irreducible over K but

geometrically reducible.Singularity: The exceptional divisor itself has only smooth or normal crossing

points, thus BsX has only cA type points. The (x1 = y1 = 0) line is singular ifs < m and generically smooth for s = m. BmX is terminal.


dy1∧dz1∧dt1x1

, so a(E, X) = 1.Divisors with discrepancy 1: Take the (1, 1, 1, 1)-blow up. BX is singular along a

line with an A2m−2 transversal section. We can blow up the line (m− 1)-times. Ateach time the exceptional divisor is a pair of transversally intersecting planes, thuswe have only cA type singularities. After (m − 1) blow ups we obtain g : Y → Xand Y has only isolated cA points, hence terminal. By (6.8), all the exceptionaldivisors over 0 ∈ X with discrepancy 1 are birational to divisors on Y . They all


72 JANOS KOLLAR

come in conjugate pairs and have been enumerated by the above (s, s, 1, 1) blowups.

Conclusion: There is a unique g–extraction whose exceptional divisor has dis-crepancy 1. It is the (m, m, 1, 1)-blow up BmX → X . Its exceptional divisor isgeometrically reducible, so we need to look further.

Divisors with discrepancy 2: Let F be a geometrically irreducible exceptionaldivisor over 0 ∈ X with discrepancy 2. Then centerBmX F is real. The centercannot be the whole (x1 = y1 = 0) line or a smooth point on it since both wouldgive a(F, X) ≥ 3. Thus it is one of the singular points, corresponding to a linearfactor of g2m+1.

By a linear change of the z, t-coordinates we may assume that this linear factoris z. Thus centerBmX F is the origin of the t-chart, where BmX has equationax2

1 + by21 + t−2m

1 g≥2m+1(z1t1, t1). This is again a cA+>1 type point (mult0 g can

be even or odd) and a(F, BmX) = 1. We have already enumerated all these cases,and we know that F is obtained by an (r, r, 1, 1)-blow up. Putting the two stepstogether, we see that F is obtained from X by an (m+ r, m+ r, 2, 1)-blow up. Nextwe compute these.

Normal form: ax2 + by2 + g≥2m+1(z, t), where m ≥ 1, −ab is not a square,g2m+1 6= 0 and mult0 g(Z2, T ) ≥ 2(m + r).

Weights for blow-up: (m + r, m + r, 2, 1), giving BrX → X .z-chart: ax2

1 + by21 + z

−2(m+r)1 g≥2m+1(z2

1 , t1z1)/ 12 (m + r, m + r, 1, 1).

Singularity: If m+r is even, then the action has a fixed curve on BrX, so BrX isnot terminal. If m+r is odd and the origin is in BrX, then we get an index 2 point.z−2(m+r)1 g≥2m+1(z2

1 , t1z1) does not vanish at the origin iff zm+r ∈ g≥2m+1(z, t).This implies that r ≥ m + 1. But g2m+1(z2

1 , t1z1) itself is not divisible by z4m+31 ,

hence r = m + 1.Conclusion: Assume that there is a linear change of the (z, t)-coordinates such

that

g≥2m+1(z, t) =∑

2i+j≥2m+2r

γijzitj and γij 6= 0 for some 2i + j = 2m + 2r.

In this coordinate system, the (m + r, m + r, 2, 1) blow up gives an elementaryextraction whose exceptional divisor is geometrically irreducible and has discrep-ancy 2. Thus the only possible g–extractions are this weighted blow up and the(m, m, 1, 1) blow up found earlier.

The (m + r, m + r, 2, 1) blow up is a g–extraction only in the r = m + 1 case:

g≥2m+1(z, t) =∑

2i+j≥4m+2

γijzitj and γ2m+1,0 6= 0.

In some cases (cf. (10.11)), we do not have any geometrically irreducible exceptionaldivisor over 0 ∈ X with discrepancy 2. Then we have to compute further withdiscrepancy 3. Fortunately, we can stop there.

Divisors with discrepancy 3:Normal form: ax2 + by2 + g≥2m+1(z, t), where m ≥ 1, −ab is not a square and

g2m+1 6= 0.Weights for blow-up: (2m + 1, 2m + 1, 2, 2), giving Y → X .z-chart: ax2

1 + by21 + z−4m−2

1 g≥2m+1(z21 , t1z

21)/ 1

2 (1, 1, 1, 0).Exceptional divisor: E := (t1 = ax2

1 +by1 +g2m+1(1, t1) = 0). It is geometricallyirreducible.



Singularity: We get an index 2 point corresponding to the linear factors ofg2m+1(z, t). Thus over a real closed field there is always an index 2 point.


1dy1∧dz1∧dt1

x1, so a(E, X) = 3.

Final conclusion: These singularities always have g–extractions. One is the(m, m, 1, 1)-blow up. Its exceptional divisor is geometrically reducible. This is theonly g–extraction with discrepancy 1.

In some cases after a suitable coordinate change we can also perform the(2m + 1, 2m + 1, 2, 1) blow up. This is the only possible g–extraction whose excep-tional divisor is geometrically irreducible and has discrepancy 2.

If g2m+1(z, t) has no linear factors over K, then the (2m + 1, 2m + 1, 2, 2) blowup is a g–extraction whose exceptional divisor is geometrically irreducible and hasdiscrepancy 3. This is the only such one. This case never happens over a real closedfield.

We have enumerated all g–extractions which either have discrepancy 1 or havea geometrically irreducible exceptional divisor. There may be other g–extractionswhose exceptional divisor is geometrically reducible and has discrepancy 2. I haveno such examples.

Example 10.11. Consider the singularity X := (x2 + y2 + zm + tn = 0) for m, nodd and m + 2 ≤ n ≤ 2m − 1. The above computations show that there is nogeometrically irreducible exceptional divisor over 0 ∈ X with discrepancy ≤ 2.

10.12 (cA+>1/2, mult0 g odd). Normal form: ax2 + by2 + g≥2m+1(z, t)/ 1

2 (1, 1, 0, 1),where −ab is not a square and g2m+1 6= 0.

Weights for blow-up: For weight reasons, only even powers of t appear in g.Thus we can define an integer r by 2m + 2r = mult0 g(Z2, T ). r ≤ m + 1 sinceg2m+1 6= 0. We consider the (s, s, 2, 1) blow up for some s ≤ m + r.

z-chart: ax21 + by2

1 + z−2s1 g≥2m+1(z2

1 , t1z1)/ 12 (s, s, 1, 1) and then we have to take

the quotient by the 12 (1, 1, 0, 1)-action. This lifts to a 1

2 (1, 1, 0, 1)-action on BX.We get a pair of commuting Z2-actions.


1 +∑

2i+j=2s γijtj1 = 0) is geometrically

irreducible.Discrepancy: π∗ dy∧dz∧dt

x = 2z21

dy1∧dz1∧dt1x1

, so a(E, X) = 2.Singularities: If s is even, then the Z2 × Z2-action is free in codimension one.

One of the elements acts by (0, 0, 1, 1), thus we get a singular curve in BX .If s is odd, then one of the elements acts by (0, 0, 1, 0). Coordinates on the

quotient are given by x1, t1, z2 = z21 , t1 and we get the equation

ax21 + by2

1 + z−s2 h≥s(z2, t

21z2)/ 1

2 (1, 1, 0, 1)

where h(Z, T 2) = g(Z, T ). At the origin we get a Z2-fixed point unless zs ∈ g.Thus s ≥ 2m + 1. On the other hand s ≤ m + r ≤ 2m + 1, thus r = m + 1 ands = m + r. Computing the t-chart shows that BX has an index 2 point unlesst4m+2 ∈ g.

Discrepancy: From this we see that a(E, X) = 1/2 if s is odd and a(E, X) = 2if s is even.

Conclusion: If m + r is odd, then a g–extraction exists iff

g≥2m+1(z, t) =∑

2i+j≥4m+2

γijzitj and γ2m+1,0 6= 0 6= γ0,4m+2.


74 JANOS KOLLAR

If this holds, then the (2m + 1, 2m + 1, 2, 1) blow up is the unique g–extraction. Ithas a geometrically irreducible exceptional divisor with discrepancy 1/2.

If m + r is even, then we have to consider g–extractions with discrepancy 1/2 or1. These can be determined by classifying all Z2-invariant divisors of discrepancy1 and pointwise Z2-fixed divisors of discrepancy 2 or 3 over X .

Divisors of discrepancy 1 over X are described in (10.10). They correspond tothe (s, s, 1, 1)-blow ups and we always get an index 2 point on the z-chart.

Let F be a pointwise Z2-fixed geometrically irreducible exceptional divisor ofdiscrepancy 2 or 3 over X. Consider the (m, m, 1, 1)-blow up.

z-chart: x21 + y2

1 + z−2m1 g≥2m+1(z1, t1z1)/ 1

2 (1, 1, 0, 1).t-chart: x2

2 + y22 + t−2m

2 g≥2m+1(z2t2, t2)/ 12 (1 −m, 1−m, 1, 1).

From this we see that the center of F on BX is the origin of one of the abovecharts. Furthermore, a(F, BX) ≤ 2. Such divisors have been described in (10.10).Thus we conclude that F is obtained by an (s, s, p, q)-blow up where p + q ≤ 4.On the z-chart of the (s, s, p, q)-blow up the Z2-action is still (1, 1, 0, 1), hence theexceptional divisor is not pointwise fixed.

The enumeration of Z2-fixed geometrically reducible divisors of discrepancy 2 or3 is harder and it seems to require separate consideration of about a dozen cases (Ihave not done all of them).

If K = R, then the situation simplifies considerably. Since mult0 g is odd, theorigin is not an isolated real point of (g = 0). So by [Kollar97b, 5.9], the linkof a cA+

>1/2 singularity with mult0 g odd always contains a connected componenthomeomorphic to RP2. By (5.4) this implies that there are no g–extractions withgeometrically reducible exceptional divisor.

Conclusion: If K = R and m + r is even, then there are no g–extractions.Probably the same result holds over any field.

11. cE-type points

In this section we study g–extractions of cE type terminal singularities. Theconventions of (8.1), (9.1) and of (9.2) are used throughout.

11.1 (cE6 main series). Normal form: x2 + y3 + yg≥3(z, t) + h≥4(z, t).Weights for blow-up: (2,2,1,1).y-chart: x2

1 + y21 + h4(z1, t1) + y1Φ(y1, z1, t1)/ 1

2 (0, 1, 1, 1).Exceptional divisor: E := (y1 = x2

1 + h4(z1, t1) = 0). E is geometrically irre-ducible iff h4 is not a square over K. If −h4 is a square over K, then E has 2geometrically irreducible components. In the other cases E is irreducible over Kbut reducible over K. Both of the components are fixed by the Z2-action, so thesame 3 cases happen for E.

Singularity: The origin is a fixed point of the Z2-action which is on BX. So weget an index 2 point on BX .

Discrepancy: π∗ dy∧dz∧dtx = 2y1

dy1∧dz1∧dt1x1

, so a(E, X) = 1.Conclusion: If −h4 is a square over K, then there are 2 geometrically irreducible

divisors with discrepancy 1, so no g–extractions. If h4 is not a square over K, thenwe get an index 2 point, so again there are no g–extractions.

11.2 (cE/2). Normal form: x2 + y3 + yg≥3(z, t)+h≥4(z, t)/ 12 (1, 0, 1, 1). By weight

considerations g3 = 0 and h5 = 0. h4 6= 0 since otherwise we would not have aterminal point. This is a cE6/2 point.



Weights for blow-up: (2,2,1,1).y-chart:

x21 + y2

1 + h4(z1, t1) + y1Φ(y1, z1, t1)/ 12 (0, 1, 1, 1).

Lifting of the Z2-action: The Z2-action lifts to 12 (1, 1, 0, 0). Thus on BX we have

two commuting Z2-actions.Exceptional divisor: E := (y1 = x2

1 + h4(z1, t1) = 0). It is geometrically irre-ducible iff h4 is not a square over K. If h4 = −Q2(z1, t1)2, then E has 2 geomet-rically irreducible components (y1 = x1 ± Q2(z1, t1) = 0). The 1

2 (1, 1, 0, 0)-actioninterchanges the 2 components, thus E ⊂ BX is geometrically irreducible.

Singularity: The 12 (1, 1, 0, 0)-action has a fixed curve, thus we get a nonterminal

singular curve on BX .Discrepancy: π∗ dy∧dz∧dt

x = 2y1dy1∧dz1∧dt1

x1, so a(E, X) = 1.

11.3 (cE7 main series). Normal form: x2 + y3 + yg≥3(z, t) + h≥5(z, t).Weights for blow-up: (3,2,1,1).x-chart:

x1 + y31x1 + y1g3(z1, t1) + h5(z1, t1) + x1Φ(y1, z1, t1)/ 1

3 (1, 1, 2, 2).

Exceptional divisor: E := (x1 = y1g3(z1, t1)+h5(z1, t1) = 0). It is geometricallyirreducible iff g3 and h5 have no common factors.

Singularity: The origin is a fixed point of the Z3-action which is on BX. Sowe get an index 3 terminal point on BX . In fact, it is the index 3 terminal pointA3/ 1

3 (1, 1, 2).Discrepancy: π∗ dy∧dz∧dt

x = 2x1 · dy1 ∧ dz1 ∧ dt1, so a(E, X) = 1.Conclusion: If g3 and h5 have no common factors, then E is irreducible and

there are no g–extractions.

11.4 (cE with common linear factors). Normal form:

x2 + y3 + yzG2(z, t) + z2Q2(z, t) + zH4(z, t) + yg≥4(z, t) + h≥6(z, t).

The following cases are of this form:cE8: x2 + y3 + yg≥4(z, t) + h≥5(z, t), if h5 has a linear factor over K, which we

can call z.cE7: x2 + y3 + yg≥3(z, t) + h≥5(z, t), if g3 and h5 have a common linear factor

over K, which we can call z.cE6: x2 + y3 + yg≥3(z, t) + h≥4(z, t), if there is a linear factor over K, which we

can call z, such that z2|h4, z|g3 and z|h5.Weights for blow-up: (3,2,2,1).z-chart:

x21 + y3

1 + y1G2(0, t1) + Q2(0, t1) + H4(0, t1)+ y1g4(0, t1) + h6(0, t1) + z1Φ(y1, z1, t1)/ 1

2 (1, 0, 1, 1).

Exceptional divisor: E is geometrically irreducible:

(z1 = x21 + y3

1 + y1G2(0, t1) + Q2(0, t1) + H4(0, t1) + y1g4(0, t1) + h6(0, t1) = 0).

Singularity: The origin is a fixed point of the Z2-action which is on BX. So weget an index 2 point on BX .


dy1∧dz1∧dt1x1

, so a(E, X) = 1.


76 JANOS KOLLAR

11.5 (cE6 with h4 a square). Normal form: x2 + y3 + yg≥3(z, t) + h≥4(z, t).Weights for blow-up: (1,1,1,1).t-chart:

x21 + y3

1t1 + y1t21g3(z1, 1) + y1t

31g4(z1, 1) + t21h4(z1, 1) + t31h5(z1, 1) + t41Φ(y1, z1, t1).

The z-chart is similar.Exceptional divisor: E := (t1 = x1 = 0), and the scheme theoretic exceptional

divisor is 2E.Singularity: On the t-chart the singular set is the line L := (x1 = y1 = t1 = 0).

We determine the singularities along this line. For a fixed value z1 = b ∈ K we geta cA-point if h4(b, 1) 6= 0. If h4(z, 1) has a simple root at b, then we get a cD-point.If h4(z, 1) has a multiple root at b, then we still get a cD point if g3(b, 1) 6= 0 anda cE-point if h5(b, 1) 6= 0.

Hence, if b ∈ K and we do not have a cDV point, then h4 has a multiple linearfactor which also divides g3 and h5. This case was settled in (11.4). Assuming thatthis is not the case, we obtain that BX has cDV points along L.

The z-chart is similar and easy computations show that the x and y-charts aresmooth along E.


dy1∧dz1∧dt1x1

, so a(E, X) = 2.First conclusion: BX is not a g–extraction since it has a singular curve. E is

geometrically irreducible and a(E, X) = 2, thus if g : Z → X is a g–extraction withexceptional divisor F , then a(F, X) = 1 by (6.11).

Computations: Here we determine all divisors F over 0 ∈ X with a(F, X) = 1.If centerBX F is not on L, then a(F, X) ≥ 3, and if centerBX F is a point on L,then a(F, X) ≥ 2. Thus if a(F, X) = 1, then centerBX F = L and a(F, BX) = 0.

Along L the threefold BX has transversal type A5 whose singularity is resolvedby blowing up the line 3 times. By explicit computation we see that only the first ofthese produces an exceptional divisor F with a(F, X) = 1. This is the same divisorthat we encountered in the (2, 2, 1, 1)-blow up and so it was already accounted for.

Final conclusion: There is no g–extraction except possibly when there is a b ∈K \ K such that (z − bt)2|h4, (z − bt)|g3, (z − bt)|h5. In these cases the samedivisibilities hold if we replace b by its conjugates over K. Thus b is quadratic overK, a root of Q2(z, 1). If F is any divisor over 0 ∈ X with a(F, X) = 1, then itscenter in BX is (z1−b = 0) ∈ L or its conjugate. Thus F is geometrically reducible.

11.6 (cE6 last case). Normal form:

x2 + y3 + cQ2(z, t)2 + yL1(z, t)Q2(z, t) + C3(z, t)Q2(z, t) + yg≥4(z, t) + h≥6(z, t),

where Q2 is a quadratic form which is irreducible over K and −c is not a square inK. By a coordinate change as in [AGV85, I.12.6] we can bring this to the simplerform

x2 + y3 + cQ2(z, t)2 + yg≥4(z, t) + h≥6(z, t),

though this is not important.Normal form and topology over R: We can choose Q2 to be positive definite and

diagonalize it. −c ∈ R is not a square, so we can choose c = 1. Thus we get thenormal form

x2 + y3 + (z2 + t2)2 + yg≥4(z, t) + h≥6(z, t).

By [Kollar97b, 4.9] we obtain that X(R) is homeomorphic to R3.



g–extractions: As we discussed above, all the g–extractions of X have geometri-cally reducible exceptional divisors.

Construction of g–extractions: It turns out that in these cases there is a g–extraction. By the above remark, we do not need to know this for certain tounderstand the topology over R, thus we only outline the construction.

Basic constructions of toric geometry are used without reference; see [Fulton93]for an introduction.

Over K we can bring the equation to the form

x2 + y3 + z2t2 + yg≥4(z, t) + h≥6(z, t).

Let ex, ey, ez, et be a basis of R4. Consider the vectors wz = 18 (3, 2, 2, 1) and

wt = 18 (3, 2, 1, 2). These vectors give a triangulation of the simplex with vertices

ex, ey, ez, et where the edges are

(ex, wz), (ex, wt), (ey, wz), (ey, wt), (ez, wz), (et, wt).

Let us take the corresponding toric blow up. One can check by a routine computa-tion that all singularities of BX are terminal and we get two index 3 points on thechart corresponding to the simplex (ex, ey, wz , wt).

Note that the above construction is symmetric in z and t. Thus if we start witha quadratic form Q2 = z2 + qt2 and introduce new coordinates z′ = z +

√qt and

t′ = z − √qt, then Q2 = z′t′ and any blow up which is symmetric in z′, t′ can be

transformed back to a blow up of X defined over K. We need to check that the twoindex 3 points become conjugates over K, but this is easy to see from the explicitequations.

11.7 (cE7 with common nonlinear factor). Normal form: x2 + y3 + yg≥3(z, t) +h≥5(z, t), where we assume that the greatest common divisor of g3 and h5 is K-irreducible (and nonconstant). We write g3 =Q(z, t)G(z, t) and h5 =Q(z, t)H(z, t).(Q is allowed to be linear, though this case is treated already.)

Weights for blow-up: (3,2,1,1).x-chart: x1 + y3

1x1 + y1g3(z1, t1) + h5(z1, t1) + x1Φ(y1, z1, t1)/ 13 (1, 1, 2, 2).

Exceptional divisor: It has two irreducible components:

E := (x1 = y1G(z1, t1) + H(z1, t1) = 0), and

F := (x1 = Q(z1, t1) = 0).

E is geometrically irreducible, F is irreducible but geometrically reducible if Q isnot linear.

Discrepancy: π∗ dy∧dz∧dtx = 3x1

dy1∧dz1∧dt1x1

, so a(E, X) = 1 = a(F, X). (Thelatter equality uses that Q is not a multiple factor.)

Further aim: We would like to construct a birational morphism g : Z → Xwhose exceptional divisor corresponds to E, and determine the singularities of Z.Thus in BX we have to contract F . F is not Q-Cartier in BX and F cannot becontracted in BX . First we have to correct this problem.

Singularities of BX : We claim that BX has only canonical singularities. Thiscan be done in 2 ways. One can compute each chart explicitly, which is rathertedious. It is easier to use a degeneration argument as follows. Let F be thenormal form of the equation as above. We may assume that g3(1, 0) = 1. Considerthe substitution

F (x, y, z, t) 7→ ε−24F (ε12x, ε8y, ε6z, ε7t).


78 JANOS KOLLAR

The exponents are chosen so that for ε → 0 the limit is X0 := (x2 + y3 + yz3 = 0).The (3, 2, 1, 1)-blow up BX0 is easy to compute. We find an index 3 terminal pointA3/ 1

3 (1, 1, 2), a curve of cA-points and a curve of cE7-points corresponding to thet-axis. Thus BX , as a small deformation of BX0, has an index 3 point at the originand some cDV singularities. (These turn out to be isolated points but we do notneed this.) As in (11.3) we see that the index 3 point is at the origin of the x-chartand it is A3/ 1

3 (1, 1, 2). In particular it is Q-factorial.Small blow-up: Let p : Y → BX be the blow up of F in BX . Let F ′ ⊂ Y denote

the birational transform of F . Away from the index 3 point BX is locally isomorphicto BX. F is defined by 2 equations (x1 = Q(z1, t1) = 0), thus p : Y → BX issmall and is an isomorphism at all points where F is Q-Cartier. The index 3 pointis Q-factorial, so F is Q-Cartier there. Thus p : F ′ → F is an isomorphism.

Contracting F ′: F is a cone over a K-irreducible curve, hence its cone ofcurves over K is 1-dimensional. If C ⊂ F ′ is a general curve, then (C · KY ) =(p(C) ·KBX) < 0 and (C · F ′) = (p(C) · F ) < 0. Thus the curves in F ′ generate aKY -negative extremal ray of Y/X , which can be contracted. We obtain f : Y → Zand g : Z → X . P := f(F ′) is a K-point since F ′ is connected.

Conclusion: g : Z → X has a geometrically irreducible exceptional divisor cor-responding to E and it has discrepancy 1. Furthermore, by (6.11) the index of Pcannot be one since a(F, X) = 1. Hence there are no g–extractions.

11.8 (Conclusion). The cE8 case is settled if h5(z, t) has a linear factor over K.This always holds if K is real closed, hence at least in this case there are no g–extractions. (I do not know what happens if K is not real closed.)

The cE7 case is settled if g3(z, t) and h5(z, t) have no common factor, or if theyhave a common linear factor over K or if they have a unique common factor overK. This accounts for all the possibilities, hence there are no g–extractions.

The cE6 case is settled if h4(z, t) is not a square over K, if −h4(z, t) is a squareover K or if h4(z, t) is divisible by the square of a linear form over K. In thesecases there are no g–extractions.

The remaining case is treated in (11.6) and a g–extraction is written down ex-plicitly. Probably this is the only g–extraction in this case. For the applications inthis paper neither its existence nor its uniqueness is crucial.

Example 11.9. Let X be the cE7 type singularity x2 + y3 + yg3(z, t) + h5(z, t),where g3 and h5 do not have a common factor. It is not hard to see that X is anisolated singular point and its (3, 2, 1, 1)-blow up has only terminal singularities.As in (11.3), the y-chart on the blow up gives the exceptional divisor

E = (g3(z, t) + h5(z, t) = 0)/ 12 (1, 1, 1, 1).

This gives examples of extremal contractions whose exceptional divisor E has aquite complicated singularity along the (z = t = 0)-line.

(1) x2 + y3 + yz3 + t5. E is singular along (z = t = 0), with a transversalsingularity type z3 + t5, that is, E8.

(2) x2 + y3 + y(z − at)(z − bt)(z − ct) + t5. E has triple self-intersection alongz = t = 0.

12. Hyperbolic 3–manifolds

The aim of this section is to show that every hyperbolic 3–manifold satisfies theconditions (1.7).



Theorem 12.1. Let M be a compact hyperbolic 3–manifold. Then M does notcontain any PL–submanifold of the following types:

(1) RP2.(2) 1–sided S1 × S1.(3) 1–sided Klein bottle.

We use two facts about hyperbolic 3–manifolds. First, that their universal coveris homeomorphic to R3. Second, that their fundamental group does not contain asubgroup isomorphic to Z2 (see, for instance, [Scott83, 4.6]).

More generally, we see how these conditions fit in the framework of Thurston’sgeometrization conjecture. This version was pointed out to me by Kapovich.

Theorem 12.2. Let M be a compact 3–manifold. Assume that M = M1 # · · · #Mk, where

(i) each Mi is aspherical, and(ii) the Seifert fibered part of the Jaco–Shalen–Johannson decomposition of Mi

is orientable.Then M does not contain any PL–submanifold of the following types:

(1) RP2.(2) 1–sided S1 × S1.(3) 1–sided Klein bottle with nonorientable neighborhood.

We consider the 3 types of submanifolds separately. Condition (1.7.1) is closelyrelated to the notion of P2-irreducibility (cf. [Hempel76, p.88]).

Lemma 12.3. Let M be a 3–manifold with universal cover M .(1) If M ∼ M1 # M2, then M contains a 2–sided RP2 iff one of the summands

does.(2) Assume that M is homeomorphic to R3. Then M does not contain an RP2

and M cannot be written as a nontrivial connected sum.

Proof. Assume that F ⊂ M is a 2–sided RP2. We may assume that F is transversalto the gluing S2. Thus C = F ∩ S2 is an embedded curve in F . Assume first thatF has a connected component C1 ⊂ C which is not null homotopic in F . Then Fis not orientable along C1, and the same holds for M along C1 since F is 2–sided.M is orientable along S2, a contradiction.

Take any connected component Ci ⊂ C such that Ci ⊂ S2 bounds a disc Di

which is disjoint from C. Ci also bounds a disc D′i in F since it is null homotopic

in F . Thus we can change the embedding RP2 → M by replacing D′i with Di

and then pushing it to one side. The new embedding is still 2–sided. Repeating ifnecessary, we eventally get an embedding which is disjoint from S2, proving (1).

RP2 cannot be embedded into R3 (cf. [Greenberg-Harper81, 27.11]), thus thepreimage of RP2 in R3 is a union of copies of S2. Fix one of these and call it N .By the Schoenflies theorem (cf. [Moise77, Sec. 17]) N bounds a 3–ball B3. Atleast one element of π1(M) maps N to itself. It cannot map the inside of N toits outside since these are not homeomorphic. If it maps B3 to itself, then by theBorsuk–Ulam theorem (cf. [Fulton95, 23.20]) we have a covering transformationwith a fixed point, a contradiction.

Assume that we have S2 ∼ N ′ ⊂ M and let S2 ∼ N ⊂ M be one of thepreimages. Then N bounds a 3–ball and so does N ′.


80 JANOS KOLLAR

In order to study the conditions (1.7.2–3) we have to distinguish two cases.

12.4 (Incompressible case). Let M be a compact 3–manifold and S ⊂ M a compact1–sided torus or Klein bottle. Assume that π1(S) ↪→ π1(M). Let ∂U be theboundary of a regular neighborhood of S. Then ∂U is a 2–sided torus or Kleinbottle and π1(∂U) ↪→ π1(S) ↪→ π1(M) is an injection. This implies that ∂U isincompressible in M (cf. [Hempel76, pp.88-89]). Thus U is one of the pieces of theJaco–Shalen–Johannson decomposition of M (cf. [Scott83, p.483]). We have to be alittle more careful since U is Seifert fibered, thus it may sit inside one of the Seifertfibered components.

The fundamental group of a hyperbolic 3–manifold does not contain a subgroupisomorphic to Z2 (see, for instance, [Scott83, 4.6]), hence the incompressible casedoes not happen for hyperbolic 3–manifolds.

12.5 (Compressible case). In this case we show that M can be written as a con-nected sum with a very special summand.

Proposition 12.6. Let M be a compact 3–manifold. Then M contains a 1–sidedtorus T such that π1(T ) → π1(M) is not an injection iff M ∼ N # (S1×S2) orM ∼ N # (S1 × RP2).

Proof. Let T ⊂ U ⊂ M be a regular neighborhood. Set V = M \ U . Then∂U = ∂V ∼ S1×S1. We know that π1(∂U) injects into π1(U). If π1(∂U) ↪→ π1(V ),then π1(∂U) ↪→ π1(U) ↪→ π1(M) by Schreier’s theorem (cf. [Lyndon-Schupp77,IV.2.6]). π1(∂U) is an index 2 subgroup of π1(T ) and π1(T ) is torsion free. Thusπ1(T ) → π1(M) is also an injection, a contradiction.

Therefore, by the Loop theorem (cf. [Hempel76, 4.2]), there is an embedding ofthe disc j : (B, ∂B) ↪→ (V, ∂V ) such that the image of j(∂B) is not contractible in∂V .

Let us cut V along j(B) to get W . The boundary of W is ∂V cut along j(∂B)(which is a cylinder) with two copies of B pasted to the ends. That is, ∂W ∼ S2.Therefore M is obtained by pasting W to a 3–manifold (with boundary) K, whichis obtained from U by attaching a 2–handle.

There are two cases corresponding to whether j(∂B) gives a primitive elementof π1(T ) ∼= Z2 (hence π1(K) ∼= Z) or is contained in 2π1(T ) (hence π1(K) ∼=Z + Z2).

Proposition 12.7. Let M be a compact 3–manifold which does not contain a 2–sided RP2. Then M contains a 1–sided Klein bottle K such that π1(K) → π1(M)is not an injection iff M ∼ N # (S1×S2) or M ∼ N # (RP3 # RP3).

Proof. Let K ⊂ U ⊂ M be a regular neighborhood and set V = M \ U . As in theproof of (12.6) we obtain an embedding of the disc j : (B, ∂B) ↪→ (V, ∂V ) such thatthe image of j(∂B) is not contractible in ∂V . We again cut V along j(B) to getW . Let ∂V ∗ denote ∂V cut along j(∂B).

There are 3 cases to consider corresponding to what ∂V ∗ is:(1) (∂V ∗ is a cylinder). Then we obtain a connected sum decomposition as in

(12.6).(2) (∂V ∗ consists of two Moebius bands). Then ∂W is two disjoint projective

planes, hence M contains a 2–sided projective plane. This cannot happenby assumption.



(3) (∂V ∗ is a Moebius band). In this case j(∂B) is 1–sided in ∂V , thus Mis not orientable along j(∂B). Then j(∂B) cannot be the boundary of anembedded disc.

Remark 12.8. So far we have excluded Seifert fiber spaces from consideration.Many Seifert fiber spaces do contain 1–sided tori or Klein bottles.

If p : M → F is a Seifert fiber space and C ⊂ F a 1–sided curve not passingthrough any critical value, then p−1(C) ⊂ M is a 1–sided torus or Klein bottle.Another example can be obtained as follows. Let x, x′ ∈ F be two points such thatthe fibers over them have multiplicity 2. Let I ⊂ F be a simple path connecting xand x′. Then p−1(I) is a 1–sided Klein bottle.

It is not hard to see that if T ⊂ M is a 1–sided torus or Klein bottle such thatp(T ) is 1–dimensional (these are called vertical), then T is obtained by one of theabove constructions.

Assume now in addition that M has a geometry modelled on H2×R (cf. [Scott83,p.459]). Then by [Johannson79, 5.6], every 1–sided torus or Klein bottle in M isisotopic to a vertical one.

This way we obtain many examples of nonorientable Seifert fiber spaces whichsatisfy the conditions (1.7).

Acknowledgments

I thank M. Bestvina, S. Gersten, M. Kapovich and G. Mikhalkin for answeringmy numerous questions about 3-manifold topology and real algebraic geometry.The existence of cE6 type points in (1.10) was established with the help of V. Alex-eev. I have received helpful comments and questions from A. Bertram, Y. Flicker,M. Fried, L. Katzarkov and B. Mazur.

Partial financial support was provided by the NSF under grant number DMS-9622394.

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Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

E-mail address: [email protected]


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